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

Percentage Accurate: 68.4% → 90.5%

Time: 5.3s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

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 - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;x + \left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{t - z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -5e-253)
     (+ x (* (- y x) (* (- z t) (/ 1.0 (- a t)))))
     (if (<= t_1 0.0)
       (+ (- (/ (* (- y x) (- z a)) t)) y)
       (+ x (* (- y x) (/ (- t z) (- t a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -5e-253) {
		tmp = x + ((y - x) * ((z - t) * (1.0 / (a - t))));
	} else if (t_1 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = x + ((y - x) * ((t - z) / (t - a)));
	}
	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 + (((y - x) * (z - t)) / (a - t))
    if (t_1 <= (-5d-253)) then
        tmp = x + ((y - x) * ((z - t) * (1.0d0 / (a - t))))
    else if (t_1 <= 0.0d0) then
        tmp = -(((y - x) * (z - a)) / t) + y
    else
        tmp = x + ((y - x) * ((t - z) / (t - a)))
    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 + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -5e-253) {
		tmp = x + ((y - x) * ((z - t) * (1.0 / (a - t))));
	} else if (t_1 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = x + ((y - x) * ((t - z) / (t - a)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_1 <= -5e-253:
		tmp = x + ((y - x) * ((z - t) * (1.0 / (a - t))))
	elif t_1 <= 0.0:
		tmp = -(((y - x) * (z - a)) / t) + y
	else:
		tmp = x + ((y - x) * ((t - z) / (t - a)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e-253)
		tmp = Float64(x + Float64(Float64(y - x) * Float64(Float64(z - t) * Float64(1.0 / Float64(a - t)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = Float64(x + Float64(Float64(y - x) * Float64(Float64(t - z) / Float64(t - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_1 <= -5e-253)
		tmp = x + ((y - x) * ((z - t) * (1.0 / (a - t))));
	elseif (t_1 <= 0.0)
		tmp = -(((y - x) * (z - a)) / t) + y;
	else
		tmp = x + ((y - x) * ((t - z) / (t - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-253], N[(x + N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] * N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -4.99999999999999971e-253

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate-rev N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -4.99999999999999971e-253 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 68.4%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.0

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

Alternative 2: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{t - a}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(y - x, t\_1, x\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t z) (- t a))) (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -5e-253)
     (fma (- y x) t_1 x)
     (if (<= t_2 0.0)
       (+ (- (/ (* (- y x) (- z a)) t)) y)
       (+ x (* (- y x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) / (t - a);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -5e-253) {
		tmp = fma((y - x), t_1, x);
	} else if (t_2 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = x + ((y - x) * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) / Float64(t - a))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -5e-253)
		tmp = fma(Float64(y - x), t_1, x);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = Float64(x + Float64(Float64(y - x) * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-253], N[(N[(y - x), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -4.99999999999999971e-253

   1. Initial program 68.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. sub-negate-rev N/A

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

      14. sub-negate-rev N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   if -4.99999999999999971e-253 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 68.4%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.0

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

   4. Applied rewrites 46.0%

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

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

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

Alternative 3: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{t - z}{t - a}, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- t z) (- t a)) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -5e-253)
     t_1
     (if (<= t_2 0.0) (+ (- (/ (* (- y x) (- z a)) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((t - z) / (t - a)), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -5e-253) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(t - z) / Float64(t - a)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -5e-253)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-253], t$95$1, If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -4.99999999999999971e-253 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. sub-negate-rev N/A

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

      14. sub-negate-rev N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   if -4.99999999999999971e-253 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 68.4%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.0

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

   4. Applied rewrites 46.0%

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

Alternative 4: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t - z}{t - a}, x\right)\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-205}:\\ \;\;\;\;\left(-\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- t z) (- t a)) x)))
   (if (<= a -1.35e-16)
     t_1
     (if (<= a 3.3e-205) (+ (- (/ (* (- y x) (- z a)) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((t - z) / (t - a)), x);
	double tmp;
	if (a <= -1.35e-16) {
		tmp = t_1;
	} else if (a <= 3.3e-205) {
		tmp = -(((y - x) * (z - a)) / t) + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(t - z) / Float64(t - a)), x)
	tmp = 0.0
	if (a <= -1.35e-16)
		tmp = t_1;
	elseif (a <= 3.3e-205)
		tmp = Float64(Float64(-Float64(Float64(Float64(y - x) * Float64(z - a)) / t)) + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.35e-16], t$95$1, If[LessEqual[a, 3.3e-205], N[((-N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]) + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e-16 or 3.2999999999999999e-205 < a

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. sub-negate-rev N/A

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

      9. sub-negate-rev N/A

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

      10. frac-2neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 68.1

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

   6. Applied rewrites 68.1%

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

   if -1.35e-16 < a < 3.2999999999999999e-205

   1. Initial program 68.4%

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

   2. Taylor expanded in t around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.0

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

   4. Applied rewrites 46.0%

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

Alternative 5: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{x}{t - a}, x\right)\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z - t}{t - a} + 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -5.8e+116)
   (fma (- z t) (/ x (- t a)) x)
   (if (<= x 1.52e+184)
     (fma y (/ (- t z) (- t a)) x)
     (* (+ (/ (- z t) (- t a)) 1.0) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -5.8e+116) {
		tmp = fma((z - t), (x / (t - a)), x);
	} else if (x <= 1.52e+184) {
		tmp = fma(y, ((t - z) / (t - a)), x);
	} else {
		tmp = (((z - t) / (t - a)) + 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -5.8e+116)
		tmp = fma(Float64(z - t), Float64(x / Float64(t - a)), x);
	elseif (x <= 1.52e+184)
		tmp = fma(y, Float64(Float64(t - z) / Float64(t - a)), x);
	else
		tmp = Float64(Float64(Float64(Float64(z - t) / Float64(t - a)) + 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -5.8e+116], N[(N[(z - t), $MachinePrecision] * N[(x / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 1.52e+184], N[(y * N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000003e116

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate-rev N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mult-flip N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 42.0

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

   8. Applied rewrites 42.0%

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

   if -5.8000000000000003e116 < x < 1.52e184

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. sub-negate-rev N/A

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

      9. sub-negate-rev N/A

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

      10. frac-2neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 68.1

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

   6. Applied rewrites 68.1%

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

   if 1.52e184 < x

   1. Initial program 68.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 43.2

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

   4. Applied rewrites 43.2%

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

Alternative 6: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{x}{t - a}, x\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z t) (/ x (- t a)) x)))
   (if (<= x -5.8e+116)
     t_1
     (if (<= x 1.1e+168) (fma y (/ (- t z) (- t a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - t), (x / (t - a)), x);
	double tmp;
	if (x <= -5.8e+116) {
		tmp = t_1;
	} else if (x <= 1.1e+168) {
		tmp = fma(y, ((t - z) / (t - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - t), Float64(x / Float64(t - a)), x)
	tmp = 0.0
	if (x <= -5.8e+116)
		tmp = t_1;
	elseif (x <= 1.1e+168)
		tmp = fma(y, Float64(Float64(t - z) / Float64(t - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * N[(x / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -5.8e+116], t$95$1, If[LessEqual[x, 1.1e+168], N[(y * N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e116 or 1.1000000000000001e168 < x

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate-rev N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mult-flip N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 42.0

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

   8. Applied rewrites 42.0%

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

   if -5.8000000000000003e116 < x < 1.1000000000000001e168

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/l* N/A

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

      8. sub-negate-rev N/A

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

      9. sub-negate-rev N/A

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

      10. frac-2neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift--.f64 68.1

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

   6. Applied rewrites 68.1%

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

Alternative 7: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{if}\;a \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- z t) a) x)))
   (if (<= a -2.8e+17)
     t_1
     (if (<= a 3.25e-125)
       (fma (- t z) (/ (- y x) t) x)
       (if (<= a 6.2e+16) (/ (* (- z t) y) (- a t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((z - t) / a), x);
	double tmp;
	if (a <= -2.8e+17) {
		tmp = t_1;
	} else if (a <= 3.25e-125) {
		tmp = fma((t - z), ((y - x) / t), x);
	} else if (a <= 6.2e+16) {
		tmp = ((z - t) * y) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(z - t) / a), x)
	tmp = 0.0
	if (a <= -2.8e+17)
		tmp = t_1;
	elseif (a <= 3.25e-125)
		tmp = fma(Float64(t - z), Float64(Float64(y - x) / t), x);
	elseif (a <= 6.2e+16)
		tmp = Float64(Float64(Float64(z - t) * y) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.8e+17], t$95$1, If[LessEqual[a, 3.25e-125], N[(N[(t - z), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 6.2e+16], N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8e17 or 6.2e16 < a

   1. Initial program 68.4%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 53.9

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

   4. Applied rewrites 53.9%

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

   if -2.8e17 < a < 3.2499999999999999e-125

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in a around 0

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

      1. frac-2neg N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 36.5

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

   6. Applied rewrites 36.5%

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

   if 3.2499999999999999e-125 < a < 6.2e16

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 39.0

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

   4. Applied rewrites 39.0%

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

Alternative 8: 60.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a - t}\\ t_2 := \mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) (- a t))) (t_2 (fma z (/ (- y x) a) x)))
   (if (<= a -1.05e+76)
     t_2
     (if (<= a -8.2e-39)
       t_1
       (if (<= a 3.25e-125)
         (fma (- t z) (/ (- y x) t) x)
         (if (<= a 1.7e+40) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / (a - t);
	double t_2 = fma(z, ((y - x) / a), x);
	double tmp;
	if (a <= -1.05e+76) {
		tmp = t_2;
	} else if (a <= -8.2e-39) {
		tmp = t_1;
	} else if (a <= 3.25e-125) {
		tmp = fma((t - z), ((y - x) / t), x);
	} else if (a <= 1.7e+40) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / Float64(a - t))
	t_2 = fma(z, Float64(Float64(y - x) / a), x)
	tmp = 0.0
	if (a <= -1.05e+76)
		tmp = t_2;
	elseif (a <= -8.2e-39)
		tmp = t_1;
	elseif (a <= 3.25e-125)
		tmp = fma(Float64(t - z), Float64(Float64(y - x) / t), x);
	elseif (a <= 1.7e+40)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.05e+76], t$95$2, If[LessEqual[a, -8.2e-39], t$95$1, If[LessEqual[a, 3.25e-125], N[(N[(t - z), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.7e+40], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.05000000000000003e76 or 1.69999999999999994e40 < a

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if -1.05000000000000003e76 < a < -8.2e-39 or 3.2499999999999999e-125 < a < 1.69999999999999994e40

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 39.0

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

   4. Applied rewrites 39.0%

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

   if -8.2e-39 < a < 3.2499999999999999e-125

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in a around 0

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

      1. frac-2neg N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 36.5

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

   6. Applied rewrites 36.5%

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

Alternative 9: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - x}{t - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y x) z) (- a t))))
   (if (<= z -4.6e+212)
     (fma z (/ (- y x) a) x)
     (if (<= z -3.4e+71)
       t_1
       (if (<= z 4.5e+94) (fma t (/ (- y x) (- t a)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - x) * z) / (a - t);
	double tmp;
	if (z <= -4.6e+212) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (z <= -3.4e+71) {
		tmp = t_1;
	} else if (z <= 4.5e+94) {
		tmp = fma(t, ((y - x) / (t - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - x) * z) / Float64(a - t))
	tmp = 0.0
	if (z <= -4.6e+212)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (z <= -3.4e+71)
		tmp = t_1;
	elseif (z <= 4.5e+94)
		tmp = fma(t, Float64(Float64(y - x) / Float64(t - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e+212], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -3.4e+71], t$95$1, If[LessEqual[z, 4.5e+94], N[(t * N[(N[(y - x), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5999999999999997e212

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if -4.5999999999999997e212 < z < -3.3999999999999998e71 or 4.49999999999999972e94 < z

   1. Initial program 68.4%

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

   2. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 38.0

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

   4. Applied rewrites 38.0%

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

   if -3.3999999999999998e71 < z < 4.49999999999999972e94

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in z around 0

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

      1. frac-2neg N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

         \[\leadsto t \cdot \frac{y - x}{t - a} + x \]

      7. frac-2neg N/A

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

      8. mul-1-neg N/A

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

      9. sub-negate-rev N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. sub-negate-rev N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 47.0

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

   6. Applied rewrites 47.0%

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

Alternative 10: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- y x) a) x)))
   (if (<= a -1.05e+76)
     t_1
     (if (<= a 1.7e+40) (/ (* (- z t) y) (- a t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((y - x) / a), x);
	double tmp;
	if (a <= -1.05e+76) {
		tmp = t_1;
	} else if (a <= 1.7e+40) {
		tmp = ((z - t) * y) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(y - x) / a), x)
	tmp = 0.0
	if (a <= -1.05e+76)
		tmp = t_1;
	elseif (a <= 1.7e+40)
		tmp = Float64(Float64(Float64(z - t) * y) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.05e+76], t$95$1, If[LessEqual[a, 1.7e+40], N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05000000000000003e76 or 1.69999999999999994e40 < a

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if -1.05000000000000003e76 < a < 1.69999999999999994e40

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 39.0

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

   4. Applied rewrites 39.0%

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

Alternative 11: 53.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right) \cdot 1\\ \mathbf{if}\;t \leq -12.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y x) 1.0))))
   (if (<= t -12.5)
     t_1
     (if (<= t 6.1e-59)
       (fma z (/ (- y x) a) x)
       (if (<= t 5.9e+79) (/ (* (- y x) z) (- a t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) * 1.0);
	double tmp;
	if (t <= -12.5) {
		tmp = t_1;
	} else if (t <= 6.1e-59) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 5.9e+79) {
		tmp = ((y - x) * z) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) * 1.0))
	tmp = 0.0
	if (t <= -12.5)
		tmp = t_1;
	elseif (t <= 6.1e-59)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 5.9e+79)
		tmp = Float64(Float64(Float64(y - x) * z) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -12.5], t$95$1, If[LessEqual[t, 6.1e-59], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 5.9e+79], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -12.5 or 5.9e79 < t

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.2%

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

   if -12.5 < t < 6.0999999999999996e-59

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if 6.0999999999999996e-59 < t < 5.9e79

   1. Initial program 68.4%

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

   2. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 38.0

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

   4. Applied rewrites 38.0%

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

Alternative 12: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right) \cdot 1\\ \mathbf{if}\;t \leq -12.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y x) 1.0))))
   (if (<= t -12.5) t_1 (if (<= t 2.25e+107) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) * 1.0);
	double tmp;
	if (t <= -12.5) {
		tmp = t_1;
	} else if (t <= 2.25e+107) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) * 1.0))
	tmp = 0.0
	if (t <= -12.5)
		tmp = t_1;
	elseif (t <= 2.25e+107)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -12.5], t$95$1, If[LessEqual[t, 2.25e+107], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -12.5 or 2.25e107 < t

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.2%

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

   if -12.5 < t < 2.25e107

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

Alternative 13: 46.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right) \cdot 1\\ \mathbf{if}\;t \leq -12.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y x) 1.0))))
   (if (<= t -12.5) t_1 (if (<= t 2.95e+103) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) * 1.0);
	double tmp;
	if (t <= -12.5) {
		tmp = t_1;
	} else if (t <= 2.95e+103) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) * 1.0))
	tmp = 0.0
	if (t <= -12.5)
		tmp = t_1;
	elseif (t <= 2.95e+103)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -12.5], t$95$1, If[LessEqual[t, 2.95e+103], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -12.5 or 2.9499999999999999e103 < t

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.2%

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

   if -12.5 < t < 2.9499999999999999e103

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 40.2

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

   7. Applied rewrites 40.2%

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

Alternative 14: 32.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - x\right) \cdot 1\\ \mathbf{if}\;t \leq -10.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-123}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+78}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y x) 1.0))))
   (if (<= t -10.8)
     t_1
     (if (<= t 4e-123) (* (- x) -1.0) (if (<= t 6.4e+78) (* y (/ z a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) * 1.0);
	double tmp;
	if (t <= -10.8) {
		tmp = t_1;
	} else if (t <= 4e-123) {
		tmp = -x * -1.0;
	} else if (t <= 6.4e+78) {
		tmp = y * (z / a);
	} 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 + ((y - x) * 1.0d0)
    if (t <= (-10.8d0)) then
        tmp = t_1
    else if (t <= 4d-123) then
        tmp = -x * (-1.0d0)
    else if (t <= 6.4d+78) then
        tmp = y * (z / a)
    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 + ((y - x) * 1.0);
	double tmp;
	if (t <= -10.8) {
		tmp = t_1;
	} else if (t <= 4e-123) {
		tmp = -x * -1.0;
	} else if (t <= 6.4e+78) {
		tmp = y * (z / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y - x) * 1.0)
	tmp = 0
	if t <= -10.8:
		tmp = t_1
	elif t <= 4e-123:
		tmp = -x * -1.0
	elif t <= 6.4e+78:
		tmp = y * (z / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) * 1.0))
	tmp = 0.0
	if (t <= -10.8)
		tmp = t_1;
	elseif (t <= 4e-123)
		tmp = Float64(Float64(-x) * -1.0);
	elseif (t <= 6.4e+78)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y - x) * 1.0);
	tmp = 0.0;
	if (t <= -10.8)
		tmp = t_1;
	elseif (t <= 4e-123)
		tmp = -x * -1.0;
	elseif (t <= 6.4e+78)
		tmp = y * (z / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -10.8], t$95$1, If[LessEqual[t, 4e-123], N[((-x) * -1.0), $MachinePrecision], If[LessEqual[t, 6.4e+78], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -10.800000000000001 or 6.39999999999999989e78 < t

   1. Initial program 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-div N/A

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

      11. sub-negate-rev N/A

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

      12. sub-negate-rev N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 84.6

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 19.2%

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

   if -10.800000000000001 < t < 4.0000000000000002e-123

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 36.9

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

   7. Applied rewrites 36.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \left(-x\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.5%

       \[\leadsto \left(-x\right) \cdot -1 \]

   if 4.0000000000000002e-123 < t < 6.39999999999999989e78

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      3. lower-*.f64 16.4

         \[\leadsto \frac{z \cdot y}{a} \]

   7. Applied rewrites 16.4%

      \[\leadsto \frac{z \cdot y}{\color{blue}{a}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

      6. lift-/.f64 18.8

         \[\leadsto y \cdot \frac{z}{a} \]

   9. Applied rewrites 18.8%

      \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 32.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+107}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z a))))
   (if (<= z -6.6e+35) t_1 (if (<= z 1.05e+107) (* (- x) -1.0) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / a);
	double tmp;
	if (z <= -6.6e+35) {
		tmp = t_1;
	} else if (z <= 1.05e+107) {
		tmp = -x * -1.0;
	} 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 = y * (z / a)
    if (z <= (-6.6d+35)) then
        tmp = t_1
    else if (z <= 1.05d+107) then
        tmp = -x * (-1.0d0)
    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 = y * (z / a);
	double tmp;
	if (z <= -6.6e+35) {
		tmp = t_1;
	} else if (z <= 1.05e+107) {
		tmp = -x * -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / a)
	tmp = 0
	if z <= -6.6e+35:
		tmp = t_1
	elif z <= 1.05e+107:
		tmp = -x * -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / a))
	tmp = 0.0
	if (z <= -6.6e+35)
		tmp = t_1;
	elseif (z <= 1.05e+107)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / a);
	tmp = 0.0;
	if (z <= -6.6e+35)
		tmp = t_1;
	elseif (z <= 1.05e+107)
		tmp = -x * -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+35], t$95$1, If[LessEqual[z, 1.05e+107], N[((-x) * -1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6000000000000003e35 or 1.05e107 < z

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      3. lower-*.f64 16.4

         \[\leadsto \frac{z \cdot y}{a} \]

   7. Applied rewrites 16.4%

      \[\leadsto \frac{z \cdot y}{\color{blue}{a}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      4. associate-/l* N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

      6. lift-/.f64 18.8

         \[\leadsto y \cdot \frac{z}{a} \]

   9. Applied rewrites 18.8%

      \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

   if -6.6000000000000003e35 < z < 1.05e107

   1. Initial program 68.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

         \[\leadsto z \cdot \frac{y - x}{a} + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 36.9

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

   7. Applied rewrites 36.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \left(-x\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.5%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot -1 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* (- x) -1.0))

C

double code(double x, double y, double z, double t, double a) {
	return -x * -1.0;
}

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 * (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -x * -1.0;
}

Python

def code(x, y, z, t, a):
	return -x * -1.0

Julia

function code(x, y, z, t, a)
	return Float64(Float64(-x) * -1.0)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -x * -1.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[((-x) * -1.0), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

2. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

      \[\leadsto z \cdot \frac{y - x}{a} + x \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 48.0

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

4. Applied rewrites 48.0%

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

5. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 36.9

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

7. Applied rewrites 36.9%

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

8. Taylor expanded in z around 0

   \[\leadsto \left(-x\right) \cdot -1 \]
9. Step-by-step derivation

10. Applied rewrites 25.5%

    \[\leadsto \left(-x\right) \cdot -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y x) (- z t)) (- a t))))