Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 85.2% → 97.5%

Time: 4.5s

Alternatives: 14

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := a \cdot z - t\\ t_2 := x - y \cdot z\\ t_3 := t - a \cdot z\\ t_4 := \frac{x}{t\_3}\\ t_5 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_1}, z, t\_4\right)\\ \mathbf{elif}\;t\_5 \leq 10^{-22}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{z}{y \cdot z - x}, a, \frac{t}{t\_2}\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_1}, y, t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a z) t))
        (t_2 (- x (* y z)))
        (t_3 (- t (* a z)))
        (t_4 (/ x t_3))
        (t_5 (/ t_2 t_3)))
   (if (<= t_5 -1e-113)
     (fma (/ y t_1) z t_4)
     (if (<= t_5 1e-22)
       (/ 1.0 (fma (/ z (- (* y z) x)) a (/ t t_2)))
       (if (<= t_5 INFINITY) (fma (/ z t_1) y t_4) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * z) - t;
	double t_2 = x - (y * z);
	double t_3 = t - (a * z);
	double t_4 = x / t_3;
	double t_5 = t_2 / t_3;
	double tmp;
	if (t_5 <= -1e-113) {
		tmp = fma((y / t_1), z, t_4);
	} else if (t_5 <= 1e-22) {
		tmp = 1.0 / fma((z / ((y * z) - x)), a, (t / t_2));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = fma((z / t_1), y, t_4);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * z) - t)
	t_2 = Float64(x - Float64(y * z))
	t_3 = Float64(t - Float64(a * z))
	t_4 = Float64(x / t_3)
	t_5 = Float64(t_2 / t_3)
	tmp = 0.0
	if (t_5 <= -1e-113)
		tmp = fma(Float64(y / t_1), z, t_4);
	elseif (t_5 <= 1e-22)
		tmp = Float64(1.0 / fma(Float64(z / Float64(Float64(y * z) - x)), a, Float64(t / t_2)));
	elseif (t_5 <= Inf)
		tmp = fma(Float64(z / t_1), y, t_4);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-113], N[(N[(y / t$95$1), $MachinePrecision] * z + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 1e-22], N[(1.0 / N[(N[(z / N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * a + N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(z / t$95$1), $MachinePrecision] * y + t$95$4), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.9999999999999998e-114

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mult-flip N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   if -9.9999999999999998e-114 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e-22

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mult-flip N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-add-rev N/A

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

      13. sub-flip N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. div-flip N/A

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

      17. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-flip N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 85.7%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 85.7%

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

   7. Applied rewrites 85.7%

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

   if 1e-22 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

Alternative 2: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + -1 \cdot \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a \cdot z - t}, z, \frac{x}{t - a \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (* -1.0 (/ x z))) a)))
   (if (<= z -4.1e+229)
     t_1
     (if (<= z 4.2e+197)
       (fma (/ y (- (* a z) t)) z (/ x (- t (* a z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (-1.0 * (x / z))) / a;
	double tmp;
	if (z <= -4.1e+229) {
		tmp = t_1;
	} else if (z <= 4.2e+197) {
		tmp = fma((y / ((a * z) - t)), z, (x / (t - (a * z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a)
	tmp = 0.0
	if (z <= -4.1e+229)
		tmp = t_1;
	elseif (z <= 4.2e+197)
		tmp = fma(Float64(y / Float64(Float64(a * z) - t)), z, Float64(x / Float64(t - Float64(a * z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + N[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -4.1e+229], t$95$1, If[LessEqual[z, 4.2e+197], N[(N[(y / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * z + N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000001e229 or 4.2000000000000001e197 < z

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   if -4.1000000000000001e229 < z < 4.2000000000000001e197

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mult-flip N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

Alternative 3: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a \cdot z - t}, y, \frac{x}{t}\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -1 \cdot \frac{x}{z}}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (fma (/ z (- (* a z) t)) y (/ x t))
     (if (<= t_2 4e+256)
       (/ (fma (- z) y x) t_1)
       (/ (+ y (* -1.0 (/ x z))) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((z / ((a * z) - t)), y, (x / t));
	} else if (t_2 <= 4e+256) {
		tmp = fma(-z, y, x) / t_1;
	} else {
		tmp = (y + (-1.0 * (x / z))) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(z / Float64(Float64(a * z) - t)), y, Float64(x / t));
	elseif (t_2 <= 4e+256)
		tmp = Float64(fma(Float64(-z), y, x) / t_1);
	else
		tmp = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y + N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+256], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(y + N[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 61.7%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000001e256

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

   3. Applied rewrites 85.2%

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

   if 4.0000000000000001e256 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

Alternative 4: 89.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / t_1) <= Inf)
		tmp = fma(Float64(z / Float64(Float64(a * z) - t)), y, Float64(x / t_1));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

Alternative 5: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := t - a \cdot z\\ \mathbf{if}\;\frac{x - y \cdot z}{t\_1} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -1 \cdot \frac{x}{z}}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))))
   (if (<= (/ (- x (* y z)) t_1) 4e+256)
     (/ (fma (- z) y x) t_1)
     (/ (+ y (* -1.0 (/ x z))) a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (((x - (y * z)) / t_1) <= 4e+256) {
		tmp = fma(-z, y, x) / t_1;
	} else {
		tmp = (y + (-1.0 * (x / z))) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / t_1) <= 4e+256)
		tmp = Float64(fma(Float64(-z), y, x) / t_1);
	else
		tmp = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 4e+256], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(y + N[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000001e256

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

   3. Applied rewrites 85.2%

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

   if 4.0000000000000001e256 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

Alternative 6: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+256}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -1 \cdot \frac{x}{z}}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* a z)))))
   (if (<= t_1 4e+256) t_1 (/ (+ y (* -1.0 (/ x z))) a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= 4e+256) {
		tmp = t_1;
	} else {
		tmp = (y + (-1.0 * (x / z))) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - (y * z)) / (t - (a * z))
    if (t_1 <= 4d+256) then
        tmp = t_1
    else
        tmp = (y + ((-1.0d0) * (x / z))) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= 4e+256) {
		tmp = t_1;
	} else {
		tmp = (y + (-1.0 * (x / z))) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (a * z))
	tmp = 0
	if t_1 <= 4e+256:
		tmp = t_1
	else:
		tmp = (y + (-1.0 * (x / z))) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_1 <= 4e+256)
		tmp = t_1;
	else
		tmp = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (a * z));
	tmp = 0.0;
	if (t_1 <= 4e+256)
		tmp = t_1;
	else
		tmp = (y + (-1.0 * (x / z))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+256], t$95$1, N[(N[(y + N[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000001e256

   1. Initial program 85.2%

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

   if 4.0000000000000001e256 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

Alternative 7: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + -1 \cdot \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{t} - z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (* -1.0 (/ x z))) a)))
   (if (<= z -2.8e+82) t_1 (if (<= z 1.5e+57) (- (/ x t) (* z (/ y t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (-1.0 * (x / z))) / a;
	double tmp;
	if (z <= -2.8e+82) {
		tmp = t_1;
	} else if (z <= 1.5e+57) {
		tmp = (x / t) - (z * (y / 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 + ((-1.0d0) * (x / z))) / a
    if (z <= (-2.8d+82)) then
        tmp = t_1
    else if (z <= 1.5d+57) then
        tmp = (x / t) - (z * (y / 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 + (-1.0 * (x / z))) / a;
	double tmp;
	if (z <= -2.8e+82) {
		tmp = t_1;
	} else if (z <= 1.5e+57) {
		tmp = (x / t) - (z * (y / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + (-1.0 * (x / z))) / a
	tmp = 0
	if z <= -2.8e+82:
		tmp = t_1
	elif z <= 1.5e+57:
		tmp = (x / t) - (z * (y / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a)
	tmp = 0.0
	if (z <= -2.8e+82)
		tmp = t_1;
	elseif (z <= 1.5e+57)
		tmp = Float64(Float64(x / t) - Float64(z * Float64(y / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + (-1.0 * (x / z))) / a;
	tmp = 0.0;
	if (z <= -2.8e+82)
		tmp = t_1;
	elseif (z <= 1.5e+57)
		tmp = (x / t) - (z * (y / 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[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.8e+82], t$95$1, If[LessEqual[z, 1.5e+57], N[(N[(x / t), $MachinePrecision] - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e82 or 1.5e57 < z

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   if -2.8e82 < z < 1.5e57

   1. Initial program 85.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 50.4%

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

   6. Applied rewrites 50.4%

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

Alternative 8: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + -1 \cdot \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (* -1.0 (/ x z))) a)))
   (if (<= z -2.75e+82) t_1 (if (<= z 1.5e+57) (/ (- x (* y z)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (-1.0 * (x / z))) / a;
	double tmp;
	if (z <= -2.75e+82) {
		tmp = t_1;
	} else if (z <= 1.5e+57) {
		tmp = (x - (y * 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 = (y + ((-1.0d0) * (x / z))) / a
    if (z <= (-2.75d+82)) then
        tmp = t_1
    else if (z <= 1.5d+57) then
        tmp = (x - (y * 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 = (y + (-1.0 * (x / z))) / a;
	double tmp;
	if (z <= -2.75e+82) {
		tmp = t_1;
	} else if (z <= 1.5e+57) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + (-1.0 * (x / z))) / a
	tmp = 0
	if z <= -2.75e+82:
		tmp = t_1
	elif z <= 1.5e+57:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + Float64(-1.0 * Float64(x / z))) / a)
	tmp = 0.0
	if (z <= -2.75e+82)
		tmp = t_1;
	elseif (z <= 1.5e+57)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + (-1.0 * (x / z))) / a;
	tmp = 0.0;
	if (z <= -2.75e+82)
		tmp = t_1;
	elseif (z <= 1.5e+57)
		tmp = (x - (y * z)) / 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[(-1.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.75e+82], t$95$1, If[LessEqual[z, 1.5e+57], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.75e82 or 1.5e57 < z

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1%

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

   6. Applied rewrites 51.1%

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

   if -2.75e82 < z < 1.5e57

   1. Initial program 85.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.3%

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

Alternative 9: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-136}:\\ \;\;\;\;\frac{y \cdot z - x}{a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e-18)
   (/ (fma (- z) y x) t)
   (if (<= t 1.3e-136) (/ (- (* y z) x) (* a z)) (/ (- x (* y z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e-18) {
		tmp = fma(-z, y, x) / t;
	} else if (t <= 1.3e-136) {
		tmp = ((y * z) - x) / (a * z);
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e-18)
		tmp = Float64(fma(Float64(-z), y, x) / t);
	elseif (t <= 1.3e-136)
		tmp = Float64(Float64(Float64(y * z) - x) / Float64(a * z));
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e-18], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 1.3e-136], N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(a * z), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.0000000000000006e-18

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 51.3%

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

   if -8.0000000000000006e-18 < t < 1.3e-136

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mult-flip N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mult-flip N/A

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift--.f64 N/A

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

      12. div-add-rev N/A

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

      13. sub-flip N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. div-flip N/A

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

      17. lower-unsound-/.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 40.5%

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

   8. Applied rewrites 40.5%

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

   if 1.3e-136 < t

   1. Initial program 85.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.3%

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

Alternative 10: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+130)
   (/ y a)
   (if (<= z 2.15e+151) (/ (fma (- z) y x) t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+130) {
		tmp = y / a;
	} else if (z <= 2.15e+151) {
		tmp = fma(-z, y, x) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+130)
		tmp = Float64(y / a);
	elseif (z <= 2.15e+151)
		tmp = Float64(fma(Float64(-z), y, x) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+130], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.15e+151], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999997e130 or 2.1499999999999999e151 < z

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

   if -5.4999999999999997e130 < z < 2.1499999999999999e151

   1. Initial program 85.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 51.3%

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

Alternative 11: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+130)
   (/ y a)
   (if (<= z 2.15e+151) (/ (- x (* y z)) t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+130) {
		tmp = y / a;
	} else if (z <= 2.15e+151) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / 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 (z <= (-5.5d+130)) then
        tmp = y / a
    else if (z <= 2.15d+151) then
        tmp = (x - (y * z)) / t
    else
        tmp = y / 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 (z <= -5.5e+130) {
		tmp = y / a;
	} else if (z <= 2.15e+151) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.5e+130:
		tmp = y / a
	elif z <= 2.15e+151:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+130)
		tmp = Float64(y / a);
	elseif (z <= 2.15e+151)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.5e+130)
		tmp = y / a;
	elseif (z <= 2.15e+151)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+130], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.15e+151], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999997e130 or 2.1499999999999999e151 < z

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

   if -5.4999999999999997e130 < z < 2.1499999999999999e151

   1. Initial program 85.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.3%

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

Alternative 12: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.5e+135)
   (/ y a)
   (if (<= z 9.5e+101) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e+135) {
		tmp = y / a;
	} else if (z <= 9.5e+101) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / 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 (z <= (-9.5d+135)) then
        tmp = y / a
    else if (z <= 9.5d+101) then
        tmp = x / (t - (a * z))
    else
        tmp = y / 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 (z <= -9.5e+135) {
		tmp = y / a;
	} else if (z <= 9.5e+101) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.5e+135:
		tmp = y / a
	elif z <= 9.5e+101:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.5e+135)
		tmp = Float64(y / a);
	elseif (z <= 9.5e+101)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.5e+135)
		tmp = y / a;
	elseif (z <= 9.5e+101)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.5e+135], N[(y / a), $MachinePrecision], If[LessEqual[z, 9.5e+101], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000004e135 or 9.4999999999999995e101 < z

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

   if -9.5000000000000004e135 < z < 9.4999999999999995e101

   1. Initial program 85.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 52.6%

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

   4. Applied rewrites 52.6%

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

Alternative 13: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -38:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -38.0) (/ y a) (if (<= z 1.8e+57) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -38.0) {
		tmp = y / a;
	} else if (z <= 1.8e+57) {
		tmp = x / t;
	} else {
		tmp = y / 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 (z <= (-38.0d0)) then
        tmp = y / a
    else if (z <= 1.8d+57) then
        tmp = x / t
    else
        tmp = y / 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 (z <= -38.0) {
		tmp = y / a;
	} else if (z <= 1.8e+57) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -38.0:
		tmp = y / a
	elif z <= 1.8e+57:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -38.0)
		tmp = Float64(y / a);
	elseif (z <= 1.8e+57)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -38.0)
		tmp = y / a;
	elseif (z <= 1.8e+57)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -38.0], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e+57], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -38 or 1.8000000000000001e57 < z

   1. Initial program 85.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 35.5%

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

   4. Applied rewrites 35.5%

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

   if -38 < z < 1.8000000000000001e57

   1. Initial program 85.2%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 35.4%

         \[\leadsto \frac{x}{\color{blue}{t}} \]

   4. Applied rewrites 35.4%

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

Alternative 14: 35.4% accurate, 3.3× speedup

Math

\[\frac{x}{t} \]

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 85.2%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x}{t}} \]
3. Step-by-step derivation

   1. lower-/.f64 35.4%

      \[\leadsto \frac{x}{\color{blue}{t}} \]

4. Applied rewrites 35.4%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64
  (/ (- x (* y z)) (- t (* a z))))