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

Percentage Accurate: 68.2% → 90.7%

Time: 5.9s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{a - z} \cdot \left(t - x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 -1e-302)
     (fma (/ (- z y) (- z a)) (- t x) x)
     (if (<= t_1 0.0)
       (- (+ t (* -1.0 (/ (* y (- t x)) z))) (* -1.0 (/ (* a (- t x)) z)))
       (- x (* (/ (- z y) (- a z)) (- t x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -1e-302) {
		tmp = fma(((z - y) / (z - a)), (t - x), x);
	} else if (t_1 <= 0.0) {
		tmp = (t + (-1.0 * ((y * (t - x)) / z))) - (-1.0 * ((a * (t - x)) / z));
	} else {
		tmp = x - (((z - y) / (a - z)) * (t - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -1e-302)
		tmp = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t + Float64(-1.0 * Float64(Float64(y * Float64(t - x)) / z))) - Float64(-1.0 * Float64(Float64(a * Float64(t - x)) / z)));
	else
		tmp = Float64(x - Float64(Float64(Float64(z - y) / Float64(a - z)) * Float64(t - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-302], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(t + N[(-1.0 * N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(a * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -9.9999999999999996e-303

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   if -9.9999999999999996e-303 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.2%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 45.9

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

   4. Applied rewrites 45.9%

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

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

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{a - z} \cdot \left(t - x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 -1e-302)
     (fma (/ (- z y) (- z a)) (- t x) x)
     (if (<= t_1 0.0)
       (+ t (* -1.0 (/ (- (* y (- t x)) (* a (- t x))) z)))
       (- x (* (/ (- z y) (- a z)) (- t x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -1e-302) {
		tmp = fma(((z - y) / (z - a)), (t - x), x);
	} else if (t_1 <= 0.0) {
		tmp = t + (-1.0 * (((y * (t - x)) - (a * (t - x))) / z));
	} else {
		tmp = x - (((z - y) / (a - z)) * (t - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_1 <= -1e-302)
		tmp = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(t + Float64(-1.0 * Float64(Float64(Float64(y * Float64(t - x)) - Float64(a * Float64(t - x))) / z)));
	else
		tmp = Float64(x - Float64(Float64(Float64(z - y) / Float64(a - z)) * Float64(t - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-302], N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t + N[(-1.0 * N[(N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -9.9999999999999996e-303

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   if -9.9999999999999996e-303 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.2%

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

   2. Taylor expanded in z around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 46.5

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

   4. Applied rewrites 46.5%

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

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

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

Alternative 3: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

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

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

   9. lower--.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. lift--.f64 N/A

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

   12. div-sub N/A

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

   13. sub-negate N/A

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

   14. div-sub N/A

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

   15. frac-2neg-rev N/A

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

   16. sub-negate-rev N/A

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

   17. lift--.f64 N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 83.7%

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

Alternative 4: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. frac-2neg N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-negate-rev N/A

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

   18. lower--.f64 83.7

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

3. Applied rewrites 83.7%

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

Alternative 5: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-206}:\\ \;\;\;\;x - \frac{z}{a - z} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x)))
   (if (<= y -3e-62)
     t_1
     (if (<= y 1.45e-206) (- x (* (/ z (- a z)) (- t x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double tmp;
	if (y <= -3e-62) {
		tmp = t_1;
	} else if (y <= 1.45e-206) {
		tmp = x - ((z / (a - z)) * (t - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	tmp = 0.0
	if (y <= -3e-62)
		tmp = t_1;
	elseif (y <= 1.45e-206)
		tmp = Float64(x - Float64(Float64(z / Float64(a - z)) * Float64(t - x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -3e-62], t$95$1, If[LessEqual[y, 1.45e-206], N[(x - N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000001e-62 or 1.4500000000000001e-206 < y

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 80.0

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

   3. Applied rewrites 80.0%

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

   if -3.0000000000000001e-62 < y < 1.4500000000000001e-206

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 46.4%

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

Alternative 6: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z - y, \frac{t}{z - a}, x\right)\\ t_2 := \left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-206}:\\ \;\;\;\;x - \frac{z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z y) (/ t (- z a)) x)) (t_2 (* (- x t) (/ y (- z a)))))
   (if (<= y -6.5e+129)
     t_2
     (if (<= y -3.6e-26)
       t_1
       (if (<= y 2.35e-206)
         (- x (* (/ z (- a z)) (- t x)))
         (if (<= y 2.5e+147) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - y), (t / (z - a)), x);
	double t_2 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_2;
	} else if (y <= -3.6e-26) {
		tmp = t_1;
	} else if (y <= 2.35e-206) {
		tmp = x - ((z / (a - z)) * (t - x));
	} else if (y <= 2.5e+147) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - y), Float64(t / Float64(z - a)), x)
	t_2 = Float64(Float64(x - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = t_2;
	elseif (y <= -3.6e-26)
		tmp = t_1;
	elseif (y <= 2.35e-206)
		tmp = Float64(x - Float64(Float64(z / Float64(a - z)) * Float64(t - x)));
	elseif (y <= 2.5e+147)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+129], t$95$2, If[LessEqual[y, -3.6e-26], t$95$1, If[LessEqual[y, 2.35e-206], N[(x - N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+147], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999995e129 or 2.5000000000000001e147 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift--.f64 N/A

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

      10. sub-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-/.f64 N/A

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

      14. sub-to-mult-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 44.0%

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

   if -6.4999999999999995e129 < y < -3.6000000000000001e-26 or 2.3499999999999999e-206 < y < 2.5000000000000001e147

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 64.2

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

   8. Applied rewrites 64.2%

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

   if -3.6000000000000001e-26 < y < 2.3499999999999999e-206

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 46.4%

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

Alternative 7: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z - y, \frac{t}{z - a}, x\right)\\ t_2 := \left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-206}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t - x, x\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z y) (/ t (- z a)) x)) (t_2 (* (- x t) (/ y (- z a)))))
   (if (<= y -6.5e+129)
     t_2
     (if (<= y -3.6e-26)
       t_1
       (if (<= y 2.35e-206)
         (fma (/ z (- z a)) (- t x) x)
         (if (<= y 2.5e+147) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - y), (t / (z - a)), x);
	double t_2 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_2;
	} else if (y <= -3.6e-26) {
		tmp = t_1;
	} else if (y <= 2.35e-206) {
		tmp = fma((z / (z - a)), (t - x), x);
	} else if (y <= 2.5e+147) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - y), Float64(t / Float64(z - a)), x)
	t_2 = Float64(Float64(x - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = t_2;
	elseif (y <= -3.6e-26)
		tmp = t_1;
	elseif (y <= 2.35e-206)
		tmp = fma(Float64(z / Float64(z - a)), Float64(t - x), x);
	elseif (y <= 2.5e+147)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+129], t$95$2, If[LessEqual[y, -3.6e-26], t$95$1, If[LessEqual[y, 2.35e-206], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.5e+147], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999995e129 or 2.5000000000000001e147 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift--.f64 N/A

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

      10. sub-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-/.f64 N/A

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

      14. sub-to-mult-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 44.0%

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

   if -6.4999999999999995e129 < y < -3.6000000000000001e-26 or 2.3499999999999999e-206 < y < 2.5000000000000001e147

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 64.2

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

   8. Applied rewrites 64.2%

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

   if -3.6000000000000001e-26 < y < 2.3499999999999999e-206

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 46.4%

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

Alternative 8: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;x - \frac{z - y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- x t) (/ y (- z a)))))
   (if (<= y -6.5e+129)
     t_1
     (if (<= y 2.4e+84) (- x (* (/ (- z y) (- a z)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_1;
	} else if (y <= 2.4e+84) {
		tmp = x - (((z - y) / (a - z)) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - t) * (y / (z - a))
    if (y <= (-6.5d+129)) then
        tmp = t_1
    else if (y <= 2.4d+84) then
        tmp = x - (((z - y) / (a - z)) * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_1;
	} else if (y <= 2.4e+84) {
		tmp = x - (((z - y) / (a - z)) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - t) * (y / (z - a))
	tmp = 0
	if y <= -6.5e+129:
		tmp = t_1
	elif y <= 2.4e+84:
		tmp = x - (((z - y) / (a - z)) * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = t_1;
	elseif (y <= 2.4e+84)
		tmp = Float64(x - Float64(Float64(Float64(z - y) / Float64(a - z)) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - t) * (y / (z - a));
	tmp = 0.0;
	if (y <= -6.5e+129)
		tmp = t_1;
	elseif (y <= 2.4e+84)
		tmp = x - (((z - y) / (a - z)) * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+129], t$95$1, If[LessEqual[y, 2.4e+84], N[(x - N[(N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999995e129 or 2.4e84 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift--.f64 N/A

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

      10. sub-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-/.f64 N/A

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

      14. sub-to-mult-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 44.0%

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

   if -6.4999999999999995e129 < y < 2.4e84

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.2%

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

Alternative 9: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- x t) (/ y (- z a)))))
   (if (<= y -6.5e+129)
     t_1
     (if (<= y 2.5e+147) (fma (- z y) (/ t (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_1;
	} else if (y <= 2.5e+147) {
		tmp = fma((z - y), (t / (z - a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = t_1;
	elseif (y <= 2.5e+147)
		tmp = fma(Float64(z - y), Float64(t / Float64(z - a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+129], t$95$1, If[LessEqual[y, 2.5e+147], N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999995e129 or 2.5000000000000001e147 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift--.f64 N/A

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

      10. sub-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-/.f64 N/A

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

      14. sub-to-mult-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 44.0%

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

   if -6.4999999999999995e129 < y < 2.5000000000000001e147

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 64.2

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

   8. Applied rewrites 64.2%

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

Alternative 10: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - y}{z - a} \cdot t\\ \mathbf{if}\;z \leq -1.55:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) (- z a)) t)))
   (if (<= z -1.55)
     t_1
     (if (<= z 2.4e+82) (fma (- t x) (/ (- y z) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / (z - a)) * t;
	double tmp;
	if (z <= -1.55) {
		tmp = t_1;
	} else if (z <= 2.4e+82) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / Float64(z - a)) * t)
	tmp = 0.0
	if (z <= -1.55)
		tmp = t_1;
	elseif (z <= 2.4e+82)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -1.55], t$95$1, If[LessEqual[z, 2.4e+82], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55000000000000004 or 2.39999999999999998e82 < z

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.7

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

   6. Applied rewrites 39.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.4

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. mul-1-neg N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lift--.f64 N/A

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

      20. lower-/.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -1.55000000000000004 < z < 2.39999999999999998e82

   1. Initial program 68.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 46.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 11: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{if}\;y \leq -6.7 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+78}:\\ \;\;\;\;x - \frac{z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- x t) (/ y (- z a)))))
   (if (<= y -6.7e+125)
     t_1
     (if (<= y 1.76e+78) (- x (* (/ z (- a z)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.7e+125) {
		tmp = t_1;
	} else if (y <= 1.76e+78) {
		tmp = x - ((z / (a - z)) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - t) * (y / (z - a))
    if (y <= (-6.7d+125)) then
        tmp = t_1
    else if (y <= 1.76d+78) then
        tmp = x - ((z / (a - z)) * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) * (y / (z - a));
	double tmp;
	if (y <= -6.7e+125) {
		tmp = t_1;
	} else if (y <= 1.76e+78) {
		tmp = x - ((z / (a - z)) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - t) * (y / (z - a))
	tmp = 0
	if y <= -6.7e+125:
		tmp = t_1
	elif y <= 1.76e+78:
		tmp = x - ((z / (a - z)) * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - t) * Float64(y / Float64(z - a)))
	tmp = 0.0
	if (y <= -6.7e+125)
		tmp = t_1;
	elseif (y <= 1.76e+78)
		tmp = Float64(x - Float64(Float64(z / Float64(a - z)) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - t) * (y / (z - a));
	tmp = 0.0;
	if (y <= -6.7e+125)
		tmp = t_1;
	elseif (y <= 1.76e+78)
		tmp = x - ((z / (a - z)) * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.7e+125], t$95$1, If[LessEqual[y, 1.76e+78], N[(x - N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7000000000000003e125 or 1.76e78 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift--.f64 N/A

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

      10. sub-to-mult-rev N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-/.f64 N/A

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

      14. sub-to-mult-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 44.0%

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

   if -6.7000000000000003e125 < y < 1.76e78

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower--.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. sub-negate N/A

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

      14. div-sub N/A

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

      15. frac-2neg-rev N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 45.1%

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

Alternative 12: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - y}{z - a} \cdot t\\ \mathbf{if}\;z \leq -1550:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) (- z a)) t)))
   (if (<= z -1550.0) t_1 (if (<= z 4.6e+75) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / (z - a)) * t;
	double tmp;
	if (z <= -1550.0) {
		tmp = t_1;
	} else if (z <= 4.6e+75) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / Float64(z - a)) * t)
	tmp = 0.0
	if (z <= -1550.0)
		tmp = t_1;
	elseif (z <= 4.6e+75)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -1550.0], t$95$1, If[LessEqual[z, 4.6e+75], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1550 or 4.5999999999999997e75 < z

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.7

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

   6. Applied rewrites 39.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.4

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. mul-1-neg N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lift--.f64 N/A

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

      20. lower-/.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -1550 < z < 4.5999999999999997e75

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 48.0

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

   6. Applied rewrites 48.0%

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

Alternative 13: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \left(z - y\right) \cdot \frac{t}{z - a}\\ \mathbf{if}\;z \leq -1550:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z y) (/ t (- z a)))))
   (if (<= z -1550.0) t_1 (if (<= z 4.6e+75) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) * (t / (z - a));
	double tmp;
	if (z <= -1550.0) {
		tmp = t_1;
	} else if (z <= 4.6e+75) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) * Float64(t / Float64(z - a)))
	tmp = 0.0
	if (z <= -1550.0)
		tmp = t_1;
	elseif (z <= 4.6e+75)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] * N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1550.0], t$95$1, If[LessEqual[z, 4.6e+75], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1550 or 4.5999999999999997e75 < z

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 83.7

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

   3. Applied rewrites 83.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 39.7

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

   6. Applied rewrites 39.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 46.4

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

   8. Applied rewrites 46.4%

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

   if -1550 < z < 4.5999999999999997e75

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. frac-2neg N/A

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

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.7

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+81}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1600:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* t (- y z)) (- a z))))
   (if (<= z -3.9e+81)
     (+ x t)
     (if (<= z -1600.0) t_1 (if (<= z 6e+82) (fma (/ y a) (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * (y - z)) / (a - z);
	double tmp;
	if (z <= -3.9e+81) {
		tmp = x + t;
	} else if (z <= -1600.0) {
		tmp = t_1;
	} else if (z <= 6e+82) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t * Float64(y - z)) / Float64(a - z))
	tmp = 0.0
	if (z <= -3.9e+81)
		tmp = Float64(x + t);
	elseif (z <= -1600.0)
		tmp = t_1;
	elseif (z <= 6e+82)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.9e+81], N[(x + t), $MachinePrecision], If[LessEqual[z, -1600.0], t$95$1, If[LessEqual[z, 6e+82], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9000000000000001e81

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 19.1

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 19.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.3%

      \[\leadsto x + t \]

   if -3.9000000000000001e81 < z < -1600 or 5.99999999999999978e82 < z

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{a - z} \]

      4. lower--.f64 39.7

         \[\leadsto \frac{t \cdot \left(y - z\right)}{a - \color{blue}{z}} \]

   4. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   if -1600 < z < 5.99999999999999978e82

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)} \]

      10. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.7

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+81}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1.75:\\ \;\;\;\;\frac{t \cdot \left(z - y\right)}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot z}{z - a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.9e+81)
   (+ x t)
   (if (<= z -1.75)
     (/ (* t (- z y)) z)
     (if (<= z 6e+82) (fma (/ y a) (- t x) x) (/ (* t z) (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e+81) {
		tmp = x + t;
	} else if (z <= -1.75) {
		tmp = (t * (z - y)) / z;
	} else if (z <= 6e+82) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = (t * z) / (z - a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e+81)
		tmp = Float64(x + t);
	elseif (z <= -1.75)
		tmp = Float64(Float64(t * Float64(z - y)) / z);
	elseif (z <= 6e+82)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(t * z) / Float64(z - a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e+81], N[(x + t), $MachinePrecision], If[LessEqual[z, -1.75], N[(N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 6e+82], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t * z), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.9000000000000001e81

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 19.1

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 19.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.3%

      \[\leadsto x + t \]

   if -3.9000000000000001e81 < z < -1.75

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)} \]

      10. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.7

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(z - y\right)}{z - a}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z - a}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z} - a} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z - a} \]

      4. lower--.f64 39.7

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z - \color{blue}{a}} \]

   6. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{t \cdot \left(z - y\right)}{z - a}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z} \]

      3. lower--.f64 26.8

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z} \]

   9. Applied rewrites 26.8%

      \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z}} \]

   if -1.75 < z < 5.99999999999999978e82

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)} \]

      10. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.7

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.0

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   if 5.99999999999999978e82 < z

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \left(t - x\right)\right)} \cdot \frac{1}{a - z} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \left(y - z\right)\right)} \cdot \frac{1}{a - z} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)} \]

      10. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      18. lower--.f64 83.7

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(z - y\right)}{z - a}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z - a}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{\color{blue}{z} - a} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z - a} \]

      4. lower--.f64 39.7

         \[\leadsto \frac{t \cdot \left(z - y\right)}{z - \color{blue}{a}} \]

   6. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{t \cdot \left(z - y\right)}{z - a}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{t \cdot z}{\color{blue}{z} - a} \]
   8. Step-by-step derivation

      1. lower-*.f64 21.2

         \[\leadsto \frac{t \cdot z}{z - a} \]

   9. Applied rewrites 21.2%

      \[\leadsto \frac{t \cdot z}{\color{blue}{z} - a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -6.5e+129)
   (/ (* y (- x t)) z)
   (if (<= y 2.4e+84) (+ x t) (* y (/ (- t x) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -6.5e+129) {
		tmp = (y * (x - t)) / z;
	} else if (y <= 2.4e+84) {
		tmp = x + t;
	} else {
		tmp = y * ((t - x) / 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) :: tmp
    if (y <= (-6.5d+129)) then
        tmp = (y * (x - t)) / z
    else if (y <= 2.4d+84) then
        tmp = x + t
    else
        tmp = y * ((t - x) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -6.5e+129) {
		tmp = (y * (x - t)) / z;
	} else if (y <= 2.4e+84) {
		tmp = x + t;
	} else {
		tmp = y * ((t - x) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -6.5e+129:
		tmp = (y * (x - t)) / z
	elif y <= 2.4e+84:
		tmp = x + t
	else:
		tmp = y * ((t - x) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = Float64(Float64(y * Float64(x - t)) / z);
	elseif (y <= 2.4e+84)
		tmp = Float64(x + t);
	else
		tmp = Float64(y * Float64(Float64(t - x) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -6.5e+129)
		tmp = (y * (x - t)) / z;
	elseif (y <= 2.4e+84)
		tmp = x + t;
	else
		tmp = y * ((t - x) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -6.5e+129], N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 2.4e+84], N[(x + t), $MachinePrecision], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999995e129

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \color{blue}{\frac{x}{a - z}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{\color{blue}{x}}{a - z}\right) \]

      4. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{\color{blue}{a - z}}\right) \]

      6. lower--.f64 42.2

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - \color{blue}{z}}\right) \]

   4. Applied rewrites 42.2%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot \color{blue}{y} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      6. sub-div N/A

         \[\leadsto \frac{t - x}{a - z} \cdot y \]

      7. lift--.f64 N/A

         \[\leadsto \frac{t - x}{a - z} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\left(t - x\right) \cdot y}{\color{blue}{a - z}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(t - x\right) \cdot y}{a - \color{blue}{z}} \]

      10. sub-to-mult-rev N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot \color{blue}{a}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot a} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot a} \]

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      18. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{\mathsf{neg}\left(\left(\frac{z}{a} - 1\right)\right)} \cdot \frac{y}{a} \]

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      3. lower--.f64 24.4

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

   9. Applied rewrites 24.4%

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]

   if -6.4999999999999995e129 < y < 2.4e84

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 19.1

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 19.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.3%

      \[\leadsto x + t \]

   if 2.4e84 < y

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \color{blue}{\frac{x}{a - z}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{\color{blue}{x}}{a - z}\right) \]

      4. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{\color{blue}{a - z}}\right) \]

      6. lower--.f64 42.2

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - \color{blue}{z}}\right) \]

   4. Applied rewrites 42.2%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto y \cdot \frac{t - x}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{t - x}{a} \]

      2. lower--.f64 25.5

         \[\leadsto y \cdot \frac{t - x}{a} \]

   7. Applied rewrites 25.5%

      \[\leadsto y \cdot \frac{t - x}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+78}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- x t)) z)))
   (if (<= y -6.5e+129) t_1 (if (<= y 1.76e+78) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (x - t)) / z;
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_1;
	} else if (y <= 1.76e+78) {
		tmp = x + t;
	} 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 * (x - t)) / z
    if (y <= (-6.5d+129)) then
        tmp = t_1
    else if (y <= 1.76d+78) then
        tmp = x + t
    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 * (x - t)) / z;
	double tmp;
	if (y <= -6.5e+129) {
		tmp = t_1;
	} else if (y <= 1.76e+78) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (x - t)) / z
	tmp = 0
	if y <= -6.5e+129:
		tmp = t_1
	elif y <= 1.76e+78:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(x - t)) / z)
	tmp = 0.0
	if (y <= -6.5e+129)
		tmp = t_1;
	elseif (y <= 1.76e+78)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (x - t)) / z;
	tmp = 0.0;
	if (y <= -6.5e+129)
		tmp = t_1;
	elseif (y <= 1.76e+78)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -6.5e+129], t$95$1, If[LessEqual[y, 1.76e+78], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999995e129 or 1.76e78 < y

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \color{blue}{\frac{x}{a - z}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{\color{blue}{x}}{a - z}\right) \]

      4. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{\color{blue}{a - z}}\right) \]

      6. lower--.f64 42.2

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - \color{blue}{z}}\right) \]

   4. Applied rewrites 42.2%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot \color{blue}{y} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \cdot y \]

      6. sub-div N/A

         \[\leadsto \frac{t - x}{a - z} \cdot y \]

      7. lift--.f64 N/A

         \[\leadsto \frac{t - x}{a - z} \cdot y \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\left(t - x\right) \cdot y}{\color{blue}{a - z}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(t - x\right) \cdot y}{a - \color{blue}{z}} \]

      10. sub-to-mult-rev N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot \color{blue}{a}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot a} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\left(t - x\right) \cdot y}{\left(1 - \frac{z}{a}\right) \cdot a} \]

      13. times-frac N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \color{blue}{\frac{y}{a}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{t - x}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      16. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{1 - \frac{z}{a}} \cdot \frac{y}{a} \]

      18. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x - t\right)\right)}{\mathsf{neg}\left(\left(\frac{z}{a} - 1\right)\right)} \cdot \frac{y}{a} \]

      19. frac-2neg-rev N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{\color{blue}{y}}{a} \]

      21. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      22. lower--.f64 N/A

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{a} \]

      23. lower-/.f64 37.8

          \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \frac{y}{\color{blue}{a}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \frac{x - t}{\frac{z}{a} - 1} \cdot \color{blue}{\frac{y}{a}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      3. lower--.f64 24.4

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

   9. Applied rewrites 24.4%

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]

   if -6.4999999999999995e129 < y < 1.76e78

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 19.1

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 19.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.3%

      \[\leadsto x + t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{+78}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a - z}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 1.76e+78) (+ x t) (/ (* t y) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.76e+78) {
		tmp = x + t;
	} else {
		tmp = (t * y) / (a - z);
	}
	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) :: tmp
    if (y <= 1.76d+78) then
        tmp = x + t
    else
        tmp = (t * y) / (a - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.76e+78) {
		tmp = x + t;
	} else {
		tmp = (t * y) / (a - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= 1.76e+78:
		tmp = x + t
	else:
		tmp = (t * y) / (a - z)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 1.76e+78)
		tmp = Float64(x + t);
	else
		tmp = Float64(Float64(t * y) / Float64(a - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 1.76e+78)
		tmp = x + t;
	else
		tmp = (t * y) / (a - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 1.76e+78], N[(x + t), $MachinePrecision], N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.76e78

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 19.1

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 19.1%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 33.3%

      \[\leadsto x + t \]

   if 1.76e78 < y

   1. Initial program 68.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \color{blue}{\frac{x}{a - z}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{\color{blue}{x}}{a - z}\right) \]

      4. lower--.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{\color{blue}{a - z}}\right) \]

      6. lower--.f64 42.2

         \[\leadsto y \cdot \left(\frac{t}{a - z} - \frac{x}{a - \color{blue}{z}}\right) \]

   4. Applied rewrites 42.2%

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a - z} - \frac{x}{a - z}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot y}{a - \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot y}{a - z} \]

      3. lower--.f64 21.6

         \[\leadsto \frac{t \cdot y}{a - z} \]

   7. Applied rewrites 21.6%

      \[\leadsto \frac{t \cdot y}{\color{blue}{a - z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 33.3% accurate, 4.8× speedup

Math

\[x + t \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x t))

C

double code(double x, double y, double z, double t, double a) {
	return x + 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 + t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + t;
}

Python

def code(x, y, z, t, a):
	return x + t

Julia

function code(x, y, z, t, a)
	return Float64(x + t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + t), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Step-by-step derivation

   1. lower--.f64 19.1

      \[\leadsto x + \left(t - \color{blue}{x}\right) \]

4. Applied rewrites 19.1%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto x + t \]
6. Step-by-step derivation

7. Applied rewrites 33.3%

   \[\leadsto x + t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))