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

Percentage Accurate: 84.9% → 93.6%

Time: 3.5s

Alternatives: 12

Speedup: 0.7×

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: 84.9% 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: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \mathsf{fma}\left(\frac{y}{a \cdot z - t}, z, \frac{x}{t\_1}\right)\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z)))
        (t_2 (fma (/ y (- (* a z) t)) z (/ x t_1)))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 5e+306)
       (/ (fma (- z) y x) t_1)
       (if (<= t_3 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = fma((y / ((a * z) - t)), z, (x / t_1));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 5e+306) {
		tmp = fma(-z, y, x) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = fma(Float64(y / Float64(Float64(a * z) - t)), z, Float64(x / t_1))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 5e+306)
		tmp = Float64(fma(Float64(-z), y, x) / t_1);
	elseif (t_3 <= Inf)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * z + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 5e+306], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 4.99999999999999993e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.99999999999999993e306

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 85.0%

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

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

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

Alternative 2: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a \cdot z - t}, z, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 5e+306)
     (/ (fma (- z) y x) t_1)
     (if (<= t_2 INFINITY) (fma (/ y (- (* a z) t)) z (/ x t)) (/ y 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 <= 5e+306) {
		tmp = fma(-z, y, x) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((y / ((a * z) - t)), z, (x / t));
	} else {
		tmp = y / 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 <= 5e+306)
		tmp = Float64(fma(Float64(-z), y, x) / t_1);
	elseif (t_2 <= Inf)
		tmp = fma(Float64(y / Float64(Float64(a * z) - t)), z, Float64(x / t));
	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]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e+306], N[(N[((-z) * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(y / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * z + N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.99999999999999993e306

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 85.0%

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

   if 4.99999999999999993e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 60.4%

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

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

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

Alternative 3: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + \frac{1}{\frac{-z}{x}}}{a}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (/ 1.0 (/ (- z) x))) a)))
   (if (<= z -2e+158)
     t_1
     (if (<= z 6e+144) (/ (fma (- z) y 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 / (-z / x))) / a;
	double tmp;
	if (z <= -2e+158) {
		tmp = t_1;
	} else if (z <= 6e+144) {
		tmp = fma(-z, y, 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(Float64(-z) / x))) / a)
	tmp = 0.0
	if (z <= -2e+158)
		tmp = t_1;
	elseif (z <= 6e+144)
		tmp = Float64(fma(Float64(-z), y, 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[((-z) / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2e+158], t$95$1, If[LessEqual[z, 6e+144], N[(N[((-z) * y + x), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999991e158 or 5.9999999999999998e144 < z

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   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%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-flip N/A

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

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

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

      8. lower-unsound-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -1.99999999999999991e158 < z < 5.9999999999999998e144

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 85.0%

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

Alternative 4: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + \frac{1}{\frac{-z}{x}}}{a}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (/ 1.0 (/ (- z) x))) a)))
   (if (<= z -2e+158)
     t_1
     (if (<= z 6e+144) (/ (- x (* y z)) (fma (- z) a t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (1.0 / (-z / x))) / a;
	double tmp;
	if (z <= -2e+158) {
		tmp = t_1;
	} else if (z <= 6e+144) {
		tmp = (x - (y * z)) / fma(-z, a, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + Float64(1.0 / Float64(Float64(-z) / x))) / a)
	tmp = 0.0
	if (z <= -2e+158)
		tmp = t_1;
	elseif (z <= 6e+144)
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-z), a, t));
	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[((-z) / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2e+158], t$95$1, If[LessEqual[z, 6e+144], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999991e158 or 5.9999999999999998e144 < z

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   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%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-flip N/A

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

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

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

      8. lower-unsound-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -1.99999999999999991e158 < z < 5.9999999999999998e144

   1. Initial program 84.9%

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

      1. remove-double-neg N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. sub-flip N/A

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

      5. distribute-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 85.0

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

   3. Applied rewrites 85.0%

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

Alternative 5: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y + \frac{1}{\frac{-z}{x}}}{a}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (+ y (/ 1.0 (/ (- z) x))) a)))
   (if (<= z -2e+158)
     t_1
     (if (<= z 6e+144) (/ (- x (* y z)) (- t (* a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (1.0 / (-z / x))) / a;
	double tmp;
	if (z <= -2e+158) {
		tmp = t_1;
	} else if (z <= 6e+144) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y + (1.0d0 / (-z / x))) / a
    if (z <= (-2d+158)) then
        tmp = t_1
    else if (z <= 6d+144) then
        tmp = (x - (y * z)) / (t - (a * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + (1.0 / (-z / x))) / a;
	double tmp;
	if (z <= -2e+158) {
		tmp = t_1;
	} else if (z <= 6e+144) {
		tmp = (x - (y * z)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + (1.0 / (-z / x))) / a
	tmp = 0
	if z <= -2e+158:
		tmp = t_1
	elif z <= 6e+144:
		tmp = (x - (y * z)) / (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(Float64(-z) / x))) / a)
	tmp = 0.0
	if (z <= -2e+158)
		tmp = t_1;
	elseif (z <= 6e+144)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + (1.0 / (-z / x))) / a;
	tmp = 0.0;
	if (z <= -2e+158)
		tmp = t_1;
	elseif (z <= 6e+144)
		tmp = (x - (y * z)) / (t - (a * z));
	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[((-z) / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2e+158], t$95$1, If[LessEqual[z, 6e+144], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999991e158 or 5.9999999999999998e144 < z

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   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%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-flip N/A

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

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

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

      8. lower-unsound-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -1.99999999999999991e158 < z < 5.9999999999999998e144

   1. Initial program 84.9%

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

Alternative 6: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.4e-35)
     t_1
     (if (<= z 5.9e-65)
       (/ x (fma (- z) a t))
       (if (<= z 2.5e+19) (- (/ x t) (* y (/ z t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.4e-35) {
		tmp = t_1;
	} else if (z <= 5.9e-65) {
		tmp = x / fma(-z, a, t);
	} else if (z <= 2.5e+19) {
		tmp = (x / t) - (y * (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -1.4e-35)
		tmp = t_1;
	elseif (z <= 5.9e-65)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (z <= 2.5e+19)
		tmp = Float64(Float64(x / t) - Float64(y * Float64(z / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -1.4e-35], t$95$1, If[LessEqual[z, 5.9e-65], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+19], N[(N[(x / t), $MachinePrecision] - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e-35 or 2.5e19 < z

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   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%

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

      1. +-commutative 51.1

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

      2. frac-2neg 51.1

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

      3. sub-negate-rev 51.1

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

      4. associate-*l/ 51.1

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

      5. div-add 51.1

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

      6. fp-cancel-sub-sign-inv 51.1

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-frac-neg2 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -1.4e-35 < z < 5.89999999999999978e-65

   1. Initial program 84.9%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 5.89999999999999978e-65 < z < 2.5e19

   1. Initial program 84.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.6%

      \[\leadsto \frac{x - y \cdot z}{\color{blue}{t}} \]
   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. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 7: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.4e-35)
     t_1
     (if (<= z 5.9e-65)
       (/ x (fma (- z) a t))
       (if (<= z 2.5e+19) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.4e-35) {
		tmp = t_1;
	} else if (z <= 5.9e-65) {
		tmp = x / fma(-z, a, t);
	} else if (z <= 2.5e+19) {
		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(x / z)) / a)
	tmp = 0.0
	if (z <= -1.4e-35)
		tmp = t_1;
	elseif (z <= 5.9e-65)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (z <= 2.5e+19)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -1.4e-35], t$95$1, If[LessEqual[z, 5.9e-65], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+19], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e-35 or 2.5e19 < z

   1. Initial program 84.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.9%

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

   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%

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

      1. +-commutative 51.1

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

      2. frac-2neg 51.1

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

      3. sub-negate-rev 51.1

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

      4. associate-*l/ 51.1

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

      5. div-add 51.1

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

      6. fp-cancel-sub-sign-inv 51.1

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-frac-neg2 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -1.4e-35 < z < 5.89999999999999978e-65

   1. Initial program 84.9%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 5.89999999999999978e-65 < z < 2.5e19

   1. Initial program 84.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.6%

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

Alternative 8: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-41)
   (/ y a)
   (if (<= z 5.9e-65)
     (/ x (fma (- z) a t))
     (if (<= z 5e+71) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 5.9e-65) {
		tmp = x / fma(-z, a, t);
	} else if (z <= 5e+71) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-41)
		tmp = Float64(y / a);
	elseif (z <= 5.9e-65)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (z <= 5e+71)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-41], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.9e-65], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+71], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e-41 or 4.99999999999999972e71 < z

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

   if -2.7e-41 < z < 5.89999999999999978e-65

   1. Initial program 84.9%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if 5.89999999999999978e-65 < z < 4.99999999999999972e71

   1. Initial program 84.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.6%

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

Alternative 9: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-41)
   (/ y a)
   (if (<= z 5.9e-65)
     (/ x (- t (* a z)))
     (if (<= z 5e+71) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 5.9e-65) {
		tmp = x / (t - (a * z));
	} else if (z <= 5e+71) {
		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 <= (-2.7d-41)) then
        tmp = y / a
    else if (z <= 5.9d-65) then
        tmp = x / (t - (a * z))
    else if (z <= 5d+71) 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 <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 5.9e-65) {
		tmp = x / (t - (a * z));
	} else if (z <= 5e+71) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-41:
		tmp = y / a
	elif z <= 5.9e-65:
		tmp = x / (t - (a * z))
	elif z <= 5e+71:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-41)
		tmp = Float64(y / a);
	elseif (z <= 5.9e-65)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 5e+71)
		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 <= -2.7e-41)
		tmp = y / a;
	elseif (z <= 5.9e-65)
		tmp = x / (t - (a * z));
	elseif (z <= 5e+71)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-41], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.9e-65], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+71], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e-41 or 4.99999999999999972e71 < z

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

   if -2.7e-41 < z < 5.89999999999999978e-65

   1. Initial program 84.9%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   if 5.89999999999999978e-65 < z < 4.99999999999999972e71

   1. Initial program 84.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 51.6%

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

Alternative 10: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-41) (/ y a) (if (<= z 6e+71) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 6e+71) {
		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 <= (-2.7d-41)) then
        tmp = y / a
    else if (z <= 6d+71) 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 <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 6e+71) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-41:
		tmp = y / a
	elif z <= 6e+71:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-41)
		tmp = Float64(y / a);
	elseif (z <= 6e+71)
		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 <= -2.7e-41)
		tmp = y / a;
	elseif (z <= 6e+71)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-41], N[(y / a), $MachinePrecision], If[LessEqual[z, 6e+71], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7e-41 or 6.00000000000000025e71 < z

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

   if -2.7e-41 < z < 6.00000000000000025e71

   1. Initial program 84.9%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

Alternative 11: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.7e-41) (/ y a) (if (<= z 1.18e+45) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 1.18e+45) {
		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 <= (-2.7d-41)) then
        tmp = y / a
    else if (z <= 1.18d+45) 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 <= -2.7e-41) {
		tmp = y / a;
	} else if (z <= 1.18e+45) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.7e-41:
		tmp = y / a
	elif z <= 1.18e+45:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.7e-41)
		tmp = Float64(y / a);
	elseif (z <= 1.18e+45)
		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 <= -2.7e-41)
		tmp = y / a;
	elseif (z <= 1.18e+45)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.7e-41], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.18e+45], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7e-41 or 1.17999999999999993e45 < z

   1. Initial program 84.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

   if -2.7e-41 < z < 1.17999999999999993e45

   1. Initial program 84.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 12: 35.3% accurate, 3.6× 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 84.9%

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

2. Taylor expanded in z around 0

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

   1. lower-/.f64 35.3

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

4. Applied rewrites 35.3%

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

Reproduce

herbie shell --seed 2025170 
(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))))