Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 68.3% → 89.9%

Time: 8.5s

Alternatives: 18

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - y}{z - a} \cdot \left(t - x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- z y) (- z a)) (- t x))))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0)
       (+ t (* -1.0 (/ (- (* y (- t x)) (* a (- t x))) z)))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((z - y) / (z - a)) * (t - x));
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + (((z - y) / (z - a)) * (t - x))
    t_2 = x + (((y - z) * (t - x)) / (a - z))
    if (t_2 <= (-1d-229)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = t + ((-1.0d0) * (((y * (t - x)) - (a * (t - x))) / z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((z - y) / (z - a)) * (t - x));
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((z - y) / (z - a)) * (t - x))
	t_2 = x + (((y - z) * (t - x)) / (a - z))
	tmp = 0
	if t_2 <= -1e-229:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(z - y) / Float64(z - a)) * Float64(t - x)))
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(t + Float64(-1.0 * Float64(Float64(Float64(y * Float64(t - x)) - Float64(a * Float64(t - x))) / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((z - y) / (z - a)) * (t - x));
	t_2 = x + (((y - z) * (t - x)) / (a - z));
	tmp = 0.0;
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(t + N[(-1.0 * N[(N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - y}{z - a} \cdot \left(t - x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- z y) (- z a)) (- t x))))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((z - y) / (z - a)) * (t - x));
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(z - y) / Float64(z - a)) * Float64(t - x)))
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z a)) (- t x) x))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (z - a)), (t - x), x);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)} \]

      10. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a - z} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(t - x\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, y - z, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(t - x\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x - t}{\mathsf{neg}\left(\left(a - z\right)\right)}}, y - z, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, y - z, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{z - a}}, y - z, x\right) \]

      15. lower--.f64 79.5

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{z - a}}, y - z, x\right) \]

   3. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{a}, z - y, x\right)\\ \mathbf{if}\;a \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) a) (- z y) x)))
   (if (<= a -5.7e-49)
     t_1
     (if (<= a 1.55e-7) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / a), (z - y), x);
	double tmp;
	if (a <= -5.7e-49) {
		tmp = t_1;
	} else if (a <= 1.55e-7) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / a), Float64(z - y), x)
	tmp = 0.0
	if (a <= -5.7e-49)
		tmp = t_1;
	elseif (a <= 1.55e-7)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / a), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.7e-49], t$95$1, If[LessEqual[a, 1.55e-7], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.7000000000000003e-49 or 1.55e-7 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}\right)}{a - z} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \left(t - x\right)}}{a - z} \]

      9. lift--.f64 N/A

         \[\leadsto x - \frac{\left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \cdot \left(t - x\right)}{a - z} \]

      10. sub-negate-rev N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right) \cdot \left(t - x\right)}}{a - z} \]

      12. lower--.f64 68.3

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{a - z}} \]

   4. Taylor expanded in z around 0

      \[\leadsto x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{\color{blue}{a}} \]
   5. Step-by-step derivation

   6. Applied rewrites 46.4%

      \[\leadsto x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{a}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{\left(z - y\right) \cdot \left(t - x\right)}{a}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(z - y\right) \cdot \left(t - x\right)}{a}\right)\right) + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\left(z - y\right) \cdot \left(t - x\right)}{a}}\right)\right) + x \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - y\right) \cdot \left(t - x\right)}}{a}\right)\right) + x \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(z - y\right) \cdot \color{blue}{\left(t - x\right)}}{a}\right)\right) + x \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - y\right) \cdot \frac{t - x}{a}}\right)\right) + x \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t - x}{a} \cdot \left(z - y\right)}\right)\right) + x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t - x}{a}\right)\right) \cdot \left(z - y\right)} + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{t - x}{a}\right), z - y, x\right)} \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(t - x\right)\right)}{a}}, z - y, x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(t - x\right)\right)}{a}}, z - y, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{a}, z - y, x\right) \]

      14. lower--.f64 51.1

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{a}, z - y, x\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - t}{a}, z - y, x\right)} \]

   if -5.7000000000000003e-49 < a < 1.55e-7

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z - a}, z - y, x\right)\\ \mathbf{if}\;a \leq -3.45 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1500000:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t (- z a)) (- z y) x)))
   (if (<= a -3.45e-40)
     t_1
     (if (<= a 1500000.0) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / (z - a)), (z - y), x);
	double tmp;
	if (a <= -3.45e-40) {
		tmp = t_1;
	} else if (a <= 1500000.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / Float64(z - a)), Float64(z - y), x)
	tmp = 0.0
	if (a <= -3.45e-40)
		tmp = t_1;
	elseif (a <= 1500000.0)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.45e-40], t$95$1, If[LessEqual[a, 1500000.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4499999999999998e-40 or 1.5e6 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 55.9%

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{t}}{a - z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot t}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a - z} + x} \]

   6. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z - a}, z - y, x\right)} \]

   if -3.4499999999999998e-40 < a < 1.5e6

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.7e-49)
   (- x (* (- x t) (/ y a)))
   (if (<= a 1.55e-7) (fma (- x t) (/ (- y a) z) t) (fma (- t x) (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.7e-49) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 1.55e-7) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.7e-49)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 1.55e-7)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.7e-49], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.55e-7], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.7000000000000003e-49

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      5. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(t - x\right) \cdot \left(\mathsf{neg}\left(\frac{y}{a}\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right) \cdot \frac{y}{a}\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      8. lift--.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right)}\right)\right) \cdot \frac{y}{a} \]

      9. sub-negate-rev N/A

         \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      10. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      11. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -5.7000000000000003e-49 < a < 1.55e-7

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   if 1.55e-7 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z - a}, z, x\right)\\ t_2 := \mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 28:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+88}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+195}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t (- z a)) z x)) (t_2 (fma (- t x) (/ a z) t)))
   (if (<= z -6.5e+242)
     t_2
     (if (<= z -8e-14)
       t_1
       (if (<= z 28.0)
         (fma (- t x) (/ y a) x)
         (if (<= z 4.3e+88)
           (* (- x t) (/ y z))
           (if (<= z 6.8e+195) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / (z - a)), z, x);
	double t_2 = fma((t - x), (a / z), t);
	double tmp;
	if (z <= -6.5e+242) {
		tmp = t_2;
	} else if (z <= -8e-14) {
		tmp = t_1;
	} else if (z <= 28.0) {
		tmp = fma((t - x), (y / a), x);
	} else if (z <= 4.3e+88) {
		tmp = (x - t) * (y / z);
	} else if (z <= 6.8e+195) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / Float64(z - a)), z, x)
	t_2 = fma(Float64(t - x), Float64(a / z), t)
	tmp = 0.0
	if (z <= -6.5e+242)
		tmp = t_2;
	elseif (z <= -8e-14)
		tmp = t_1;
	elseif (z <= 28.0)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	elseif (z <= 4.3e+88)
		tmp = Float64(Float64(x - t) * Float64(y / z));
	elseif (z <= 6.8e+195)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6.5e+242], t$95$2, If[LessEqual[z, -8e-14], t$95$1, If[LessEqual[z, 28.0], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.3e+88], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+195], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.49999999999999992e242 or 6.80000000000000021e195 < z

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 32.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{z}, t\right) \]

   9. Applied rewrites 32.6%

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]

   if -6.49999999999999992e242 < z < -7.99999999999999999e-14 or 4.29999999999999974e88 < z < 6.80000000000000021e195

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 55.9%

      \[\leadsto x + \frac{\left(y - z\right) \cdot \color{blue}{t}}{a - z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot t}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a - z} + x} \]

   6. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z - a}, z - y, x\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{t}{z - a}, \color{blue}{z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(\frac{t}{z - a}, \color{blue}{z}, x\right) \]

   if -7.99999999999999999e-14 < z < 28

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   if 28 < z < 4.29999999999999974e88

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       6. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(t - x\right) \cdot y\right) \cdot \frac{1}{z}\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(t - x\right) \cdot \left(y \cdot \frac{1}{z}\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       12. lower--.f64 N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       13. mult-flip-rev N/A

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

       14. lower-/.f64 26.2

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

   11. Applied rewrites 26.2%

       \[\leadsto \left(x - t\right) \cdot \frac{y}{\color{blue}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0016:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-147}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;t + \frac{a \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.0016)
   (- x (* (- x t) (/ y a)))
   (if (<= a 9e-147)
     (/ (* y (- t x)) (- a z))
     (if (<= a 8.2e-29) (+ t (/ (* a (- t x)) z)) (fma (- t x) (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.0016) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 9e-147) {
		tmp = (y * (t - x)) / (a - z);
	} else if (a <= 8.2e-29) {
		tmp = t + ((a * (t - x)) / z);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.0016)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 9e-147)
		tmp = Float64(Float64(y * Float64(t - x)) / Float64(a - z));
	elseif (a <= 8.2e-29)
		tmp = Float64(t + Float64(Float64(a * Float64(t - x)) / z));
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.0016], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-147], N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.2e-29], N[(t + N[(N[(a * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -0.00160000000000000008

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      5. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(t - x\right) \cdot \left(\mathsf{neg}\left(\frac{y}{a}\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right) \cdot \frac{y}{a}\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      8. lift--.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right)}\right)\right) \cdot \frac{y}{a} \]

      9. sub-negate-rev N/A

         \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      10. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      11. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -0.00160000000000000008 < a < 8.99999999999999946e-147

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}\right)}{a - z} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \left(t - x\right)}}{a - z} \]

      9. lift--.f64 N/A

         \[\leadsto x - \frac{\left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \cdot \left(t - x\right)}{a - z} \]

      10. sub-negate-rev N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right) \cdot \left(t - x\right)}}{a - z} \]

      12. lower--.f64 68.3

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{a - z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto x - \color{blue}{x \cdot \left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - x \cdot \color{blue}{\left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{z - y}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a} - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - \color{blue}{z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      8. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      10. lower--.f64 64.3

          \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

   6. Applied rewrites 64.3%

      \[\leadsto x - \color{blue}{x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - z} \]

      4. lower--.f64 37.5

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - \color{blue}{z}} \]

   9. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]

   if 8.99999999999999946e-147 < a < 8.1999999999999996e-29

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

   5. Taylor expanded in y around 0

      \[\leadsto t + \frac{a \cdot \left(t - x\right)}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto t + \frac{a \cdot \left(t - x\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto t + \frac{a \cdot \left(t - x\right)}{z} \]

      3. lower--.f64 29.9

         \[\leadsto t + \frac{a \cdot \left(t - x\right)}{z} \]

   7. Applied rewrites 29.9%

      \[\leadsto t + \frac{a \cdot \left(t - x\right)}{\color{blue}{z}} \]

   if 8.1999999999999996e-29 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0016:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-147}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.0016)
   (- x (* (- x t) (/ y a)))
   (if (<= a 9e-147)
     (/ (* y (- t x)) (- a z))
     (if (<= a 8.2e-29) (fma (- t x) (/ a z) t) (fma (- t x) (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.0016) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 9e-147) {
		tmp = (y * (t - x)) / (a - z);
	} else if (a <= 8.2e-29) {
		tmp = fma((t - x), (a / z), t);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.0016)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 9e-147)
		tmp = Float64(Float64(y * Float64(t - x)) / Float64(a - z));
	elseif (a <= 8.2e-29)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.0016], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-147], N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.2e-29], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -0.00160000000000000008

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \cdot \left(t - x\right)} \]

      5. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(t - x\right) \cdot \left(\mathsf{neg}\left(\frac{y}{a}\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right) \cdot \frac{y}{a}\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      8. lift--.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right)}\right)\right) \cdot \frac{y}{a} \]

      9. sub-negate-rev N/A

         \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      10. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      11. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -0.00160000000000000008 < a < 8.99999999999999946e-147

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}\right)}{a - z} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \left(t - x\right)}}{a - z} \]

      9. lift--.f64 N/A

         \[\leadsto x - \frac{\left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \cdot \left(t - x\right)}{a - z} \]

      10. sub-negate-rev N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right) \cdot \left(t - x\right)}}{a - z} \]

      12. lower--.f64 68.3

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{a - z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto x - \color{blue}{x \cdot \left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - x \cdot \color{blue}{\left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{z - y}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a} - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - \color{blue}{z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      8. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      10. lower--.f64 64.3

          \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

   6. Applied rewrites 64.3%

      \[\leadsto x - \color{blue}{x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - z} \]

      4. lower--.f64 37.5

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - \color{blue}{z}} \]

   9. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]

   if 8.99999999999999946e-147 < a < 8.1999999999999996e-29

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 32.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{z}, t\right) \]

   9. Applied rewrites 32.6%

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]

   if 8.1999999999999996e-29 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -0.0016:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-147}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ y a) x)))
   (if (<= a -0.0016)
     t_1
     (if (<= a 9e-147)
       (/ (* y (- t x)) (- a z))
       (if (<= a 8.2e-29) (fma (- t x) (/ a z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (y / a), x);
	double tmp;
	if (a <= -0.0016) {
		tmp = t_1;
	} else if (a <= 9e-147) {
		tmp = (y * (t - x)) / (a - z);
	} else if (a <= 8.2e-29) {
		tmp = fma((t - x), (a / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(y / a), x)
	tmp = 0.0
	if (a <= -0.0016)
		tmp = t_1;
	elseif (a <= 9e-147)
		tmp = Float64(Float64(y * Float64(t - x)) / Float64(a - z));
	elseif (a <= 8.2e-29)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -0.0016], t$95$1, If[LessEqual[a, 9e-147], N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.2e-29], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.00160000000000000008 or 8.1999999999999996e-29 < a

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   if -0.00160000000000000008 < a < 8.99999999999999946e-147

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right) \cdot \left(t - x\right)\right)}{a - z}} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}\right)}{a - z} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \left(t - x\right)}}{a - z} \]

      9. lift--.f64 N/A

         \[\leadsto x - \frac{\left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \cdot \left(t - x\right)}{a - z} \]

      10. sub-negate-rev N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right) \cdot \left(t - x\right)}}{a - z} \]

      12. lower--.f64 68.3

          \[\leadsto x - \frac{\color{blue}{\left(z - y\right)} \cdot \left(t - x\right)}{a - z} \]

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{x - \frac{\left(z - y\right) \cdot \left(t - x\right)}{a - z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto x - \color{blue}{x \cdot \left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - x \cdot \color{blue}{\left(-1 \cdot \frac{z - y}{a - z} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{z - y}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a - z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{a} - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - \color{blue}{z}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      8. lower--.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

      10. lower--.f64 64.3

          \[\leadsto x - x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right) \]

   6. Applied rewrites 64.3%

      \[\leadsto x - \color{blue}{x \cdot \mathsf{fma}\left(-1, \frac{z - y}{a - z}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(a - z\right)}\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - z} \]

      4. lower--.f64 37.5

         \[\leadsto \frac{y \cdot \left(t - x\right)}{a - \color{blue}{z}} \]

   9. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]

   if 8.99999999999999946e-147 < a < 8.1999999999999996e-29

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 32.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{z}, t\right) \]

   9. Applied rewrites 32.6%

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+58}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 28:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+192}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ a z) t)))
   (if (<= z -1.3e+114)
     t_1
     (if (<= z -1.65e+58)
       (* (/ (- x t) z) y)
       (if (<= z 28.0)
         (fma (- t x) (/ y a) x)
         (if (<= z 5.4e+192) (* (- x t) (/ y z)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (a / z), t);
	double tmp;
	if (z <= -1.3e+114) {
		tmp = t_1;
	} else if (z <= -1.65e+58) {
		tmp = ((x - t) / z) * y;
	} else if (z <= 28.0) {
		tmp = fma((t - x), (y / a), x);
	} else if (z <= 5.4e+192) {
		tmp = (x - t) * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(a / z), t)
	tmp = 0.0
	if (z <= -1.3e+114)
		tmp = t_1;
	elseif (z <= -1.65e+58)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (z <= 28.0)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	elseif (z <= 5.4e+192)
		tmp = Float64(Float64(x - t) * Float64(y / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.3e+114], t$95$1, If[LessEqual[z, -1.65e+58], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 28.0], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.4e+192], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.3e114 or 5.39999999999999979e192 < z

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 32.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{z}, t\right) \]

   9. Applied rewrites 32.6%

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]

   if -1.3e114 < z < -1.64999999999999991e58

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       5. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(y \cdot \frac{t - x}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{t - x}{z} \cdot y\right) \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot y \]

       9. distribute-neg-frac2 N/A

          \[\leadsto \frac{t - x}{\mathsf{neg}\left(z\right)} \cdot y \]

       10. mult-flip-rev N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       12. frac-2neg N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{-1}{z}\right) \cdot y \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{-1}{z}\right) \cdot y \]

       14. frac-2neg N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       15. metadata-eval N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       16. mult-flip-rev N/A

           \[\leadsto \frac{t - x}{\mathsf{neg}\left(z\right)} \cdot y \]

       17. distribute-neg-frac2 N/A

           \[\leadsto \left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot y \]

       18. distribute-neg-frac N/A

           \[\leadsto \frac{\mathsf{neg}\left(\left(t - x\right)\right)}{z} \cdot y \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\mathsf{neg}\left(\left(t - x\right)\right)}{z} \cdot y \]

       20. sub-negate-rev N/A

           \[\leadsto \frac{x - t}{z} \cdot y \]

       21. lower--.f64 25.3

           \[\leadsto \frac{x - t}{z} \cdot y \]

   11. Applied rewrites 25.3%

       \[\leadsto \frac{x - t}{z} \cdot \color{blue}{y} \]

   if -1.64999999999999991e58 < z < 28

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} \]

      4. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} \]

      8. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      9. frac-2neg N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      10. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      11. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      13. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      14. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      15. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      16. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto x + \frac{y}{\color{blue}{a}} \cdot \left(t - x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \left(t - x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y}{a}} + x \]

      5. lower-fma.f64 48.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)} \]

   if 28 < z < 5.39999999999999979e192

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       6. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(t - x\right) \cdot y\right) \cdot \frac{1}{z}\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(t - x\right) \cdot \left(y \cdot \frac{1}{z}\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       12. lower--.f64 N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       13. mult-flip-rev N/A

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

       14. lower-/.f64 26.2

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

   11. Applied rewrites 26.2%

       \[\leadsto \left(x - t\right) \cdot \frac{y}{\color{blue}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.75e-63)
   (* (- x t) (/ y z))
   (if (<= y 2e+137) (fma (- t x) (/ a z) t) (* (/ (- x t) z) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.75e-63) {
		tmp = (x - t) * (y / z);
	} else if (y <= 2e+137) {
		tmp = fma((t - x), (a / z), t);
	} else {
		tmp = ((x - t) / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.75e-63)
		tmp = Float64(Float64(x - t) * Float64(y / z));
	elseif (y <= 2e+137)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	else
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.75e-63], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+137], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75000000000000002e-63

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       6. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(t - x\right) \cdot y\right) \cdot \frac{1}{z}\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(t - x\right) \cdot \left(y \cdot \frac{1}{z}\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       12. lower--.f64 N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       13. mult-flip-rev N/A

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

       14. lower-/.f64 26.2

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

   11. Applied rewrites 26.2%

       \[\leadsto \left(x - t\right) \cdot \frac{y}{\color{blue}{z}} \]

   if -1.75000000000000002e-63 < y < 2.0000000000000001e137

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 32.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{z}, t\right) \]

   9. Applied rewrites 32.6%

      \[\leadsto \mathsf{fma}\left(t - x, \frac{a}{\color{blue}{z}}, t\right) \]

   if 2.0000000000000001e137 < y

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       5. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(y \cdot \frac{t - x}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{t - x}{z} \cdot y\right) \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot y \]

       9. distribute-neg-frac2 N/A

          \[\leadsto \frac{t - x}{\mathsf{neg}\left(z\right)} \cdot y \]

       10. mult-flip-rev N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       11. metadata-eval N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       12. frac-2neg N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{-1}{z}\right) \cdot y \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{-1}{z}\right) \cdot y \]

       14. frac-2neg N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       15. metadata-eval N/A

           \[\leadsto \left(\left(t - x\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) \cdot y \]

       16. mult-flip-rev N/A

           \[\leadsto \frac{t - x}{\mathsf{neg}\left(z\right)} \cdot y \]

       17. distribute-neg-frac2 N/A

           \[\leadsto \left(\mathsf{neg}\left(\frac{t - x}{z}\right)\right) \cdot y \]

       18. distribute-neg-frac N/A

           \[\leadsto \frac{\mathsf{neg}\left(\left(t - x\right)\right)}{z} \cdot y \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\mathsf{neg}\left(\left(t - x\right)\right)}{z} \cdot y \]

       20. sub-negate-rev N/A

           \[\leadsto \frac{x - t}{z} \cdot y \]

       21. lower--.f64 25.3

           \[\leadsto \frac{x - t}{z} \cdot y \]

   11. Applied rewrites 25.3%

       \[\leadsto \frac{x - t}{z} \cdot \color{blue}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 34.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-59}:\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- x t) (/ y z))))
   (if (<= y -1.6e-59) t_1 (if (<= y 9.8e-59) (+ x (- t x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / z);
	double tmp;
	if (y <= -1.6e-59) {
		tmp = t_1;
	} else if (y <= 9.8e-59) {
		tmp = x + (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - t) * (y / z)
    if (y <= (-1.6d-59)) then
        tmp = t_1
    else if (y <= 9.8d-59) then
        tmp = x + (t - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / z);
	double tmp;
	if (y <= -1.6e-59) {
		tmp = t_1;
	} else if (y <= 9.8e-59) {
		tmp = x + (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - t) * (y / z)
	tmp = 0
	if y <= -1.6e-59:
		tmp = t_1
	elif y <= 9.8e-59:
		tmp = x + (t - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - t) * Float64(y / z))
	tmp = 0.0
	if (y <= -1.6e-59)
		tmp = t_1;
	elseif (y <= 9.8e-59)
		tmp = Float64(x + Float64(t - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - t) * (y / z);
	tmp = 0.0;
	if (y <= -1.6e-59)
		tmp = t_1;
	elseif (y <= 9.8e-59)
		tmp = x + (t - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e-59], t$95$1, If[LessEqual[y, 9.8e-59], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-59 or 9.79999999999999954e-59 < y

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right) \]

       4. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       6. lift--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\left(t - x\right) \cdot y\right) \cdot \frac{1}{z}\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(t - x\right) \cdot \left(y \cdot \frac{1}{z}\right)\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(y \cdot \color{blue}{\frac{1}{z}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       12. lower--.f64 N/A

           \[\leadsto \left(x - t\right) \cdot \left(y \cdot \frac{\color{blue}{1}}{z}\right) \]

       13. mult-flip-rev N/A

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

       14. lower-/.f64 26.2

           \[\leadsto \left(x - t\right) \cdot \frac{y}{z} \]

   11. Applied rewrites 26.2%

       \[\leadsto \left(x - t\right) \cdot \frac{y}{\color{blue}{z}} \]

   if -1.6e-59 < y < 9.79999999999999954e-59

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 28.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= t -1.35e+25) t_1 (if (<= t 7.5e-33) (/ (* x (- y a)) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (t <= -1.35e+25) {
		tmp = t_1;
	} else if (t <= 7.5e-33) {
		tmp = (x * (y - a)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (t - x)
    if (t <= (-1.35d+25)) then
        tmp = t_1
    else if (t <= 7.5d-33) then
        tmp = (x * (y - a)) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (t <= -1.35e+25) {
		tmp = t_1;
	} else if (t <= 7.5e-33) {
		tmp = (x * (y - a)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (t - x)
	tmp = 0
	if t <= -1.35e+25:
		tmp = t_1
	elif t <= 7.5e-33:
		tmp = (x * (y - a)) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (t <= -1.35e+25)
		tmp = t_1;
	elseif (t <= 7.5e-33)
		tmp = Float64(Float64(x * Float64(y - a)) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (t - x);
	tmp = 0.0;
	if (t <= -1.35e+25)
		tmp = t_1;
	elseif (t <= 7.5e-33)
		tmp = (x * (y - a)) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.35e+25], t$95$1, If[LessEqual[t, 7.5e-33], N[(N[(x * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e25 or 7.5000000000000001e-33 < t

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   if -1.35e25 < t < 7.5000000000000001e-33

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      3. lower--.f64 20.2

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(t - x\right)\\ \mathbf{if}\;t \leq -7.9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (- t x))))
   (if (<= t -7.9) t_1 (if (<= t 7.5e-33) (/ (* x y) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (t <= -7.9) {
		tmp = t_1;
	} else if (t <= 7.5e-33) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (t - x)
    if (t <= (-7.9d0)) then
        tmp = t_1
    else if (t <= 7.5d-33) then
        tmp = (x * y) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (t - x);
	double tmp;
	if (t <= -7.9) {
		tmp = t_1;
	} else if (t <= 7.5e-33) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (t - x)
	tmp = 0
	if t <= -7.9:
		tmp = t_1
	elif t <= 7.5e-33:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(t - x))
	tmp = 0.0
	if (t <= -7.9)
		tmp = t_1;
	elseif (t <= 7.5e-33)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (t - x);
	tmp = 0.0;
	if (t <= -7.9)
		tmp = t_1;
	elseif (t <= 7.5e-33)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.9], t$95$1, If[LessEqual[t, 7.5e-33], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9000000000000004 or 7.5000000000000001e-33 < t

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   if -7.9000000000000004 < t < 7.5000000000000001e-33

   1. Initial program 68.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\mathsf{neg}\left(z\right)} + t \]

      7. mult-flip N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      8. lift--.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      10. lift-*.f64 N/A

          \[\leadsto \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)} + t \]

      12. associate-*l* N/A

          \[\leadsto \left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}\right) + t \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(z\right)}}, t\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z\right)}, t\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{z}\right)}, t\right) \]

      17. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

      18. lower-/.f64 54.2

          \[\leadsto \mathsf{fma}\left(t - x, \left(y - a\right) \cdot \frac{-1}{\color{blue}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\left(y - a\right) \cdot \frac{-1}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{\color{blue}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

      4. lower--.f64 23.5

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right)}{z}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{x \cdot y}{z} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot y}{z} \]

       2. lower-*.f64 16.7

          \[\leadsto \frac{x \cdot y}{z} \]

   12. Applied rewrites 16.7%

       \[\leadsto \frac{x \cdot y}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 20.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x + \left(t - x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (- t x)))

C

double code(double x, double y, double z, double t, double a) {
	return x + (t - x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (t - x)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (t - x);
}

Python

def code(x, y, z, t, a):
	return x + (t - x)

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(t - x))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (t - x);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Step-by-step derivation

   1. lower--.f64 20.2

      \[\leadsto x + \left(t - \color{blue}{x}\right) \]

4. Applied rewrites 20.2%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
5. Add Preprocessing

Alternative 18: 2.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) + x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ (- x) x))

C

double code(double x, double y, double z, double t, double a) {
	return -x + x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -x + x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -x + x;
}

Python

def code(x, y, z, t, a):
	return -x + x

Julia

function code(x, y, z, t, a)
	return Float64(Float64(-x) + x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -x + x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[((-x) + x), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Step-by-step derivation

   1. lower--.f64 20.2

      \[\leadsto x + \left(t - \color{blue}{x}\right) \]

4. Applied rewrites 20.2%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
6. Step-by-step derivation

   1. lower-*.f64 2.8

      \[\leadsto x + -1 \cdot x \]

7. Applied rewrites 2.8%

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + -1 \cdot x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{-1 \cdot x + x} \]

   3. lower-+.f64 2.8

      \[\leadsto \color{blue}{-1 \cdot x + x} \]

   4. lift-*.f64 N/A

      \[\leadsto -1 \cdot x + x \]

   5. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + x \]

   6. lower-neg.f64 2.8

      \[\leadsto \left(-x\right) + x \]

9. Applied rewrites 2.8%

   \[\leadsto \color{blue}{\left(-x\right) + x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))