Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Percentage Accurate: 94.2% → 98.1%

Time: 9.2s

Alternatives: 15

Speedup: 0.9×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

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

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * ((y / z) - (t / (1.0d0 - z)))
end function

Java

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

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+306)))
     (* y (/ x z))
     (* x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+306)) {
		tmp = y * (x / z);
	} else {
		tmp = x * t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+306)) {
		tmp = y * (x / z);
	} else {
		tmp = x * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+306):
		tmp = y * (x / z)
	else:
		tmp = x * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+306))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+306)))
		tmp = y * (x / z);
	else
		tmp = x * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+306]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 2.00000000000000003e306 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 64.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 2.00000000000000003e306

   1. Initial program 97.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t + y}{z}\\ \mathbf{if}\;z \leq -1.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-199}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), -t, \frac{y}{z} \cdot x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ t y) z))))
   (if (<= z -1.7)
     t_1
     (if (<= z -1.15e-199)
       (fma (fma z x x) (- t) (* (/ y z) x))
       (if (<= z 1.0) (/ (* x (- y (* t z))) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((t + y) / z);
	double tmp;
	if (z <= -1.7) {
		tmp = t_1;
	} else if (z <= -1.15e-199) {
		tmp = fma(fma(z, x, x), -t, ((y / z) * x));
	} else if (z <= 1.0) {
		tmp = (x * (y - (t * z))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(t + y) / z))
	tmp = 0.0
	if (z <= -1.7)
		tmp = t_1;
	elseif (z <= -1.15e-199)
		tmp = fma(fma(z, x, x), Float64(-t), Float64(Float64(y / z) * x));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x * Float64(y - Float64(t * z))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7], t$95$1, If[LessEqual[z, -1.15e-199], N[(N[(z * x + x), $MachinePrecision] * (-t) + N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999996 or 1 < z

   1. Initial program 95.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 90.6

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 94.3

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

   11. Applied rewrites 94.3%

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

   if -1.69999999999999996 < z < -1.1500000000000001e-199

   1. Initial program 99.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. div-add N/A

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

   5. Applied rewrites 97.9%

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

   if -1.1500000000000001e-199 < z < 1

   1. Initial program 86.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. fp-cancel-sub-sign N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. fp-cancel-sub-sign N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

Alternative 3: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t + y}{z}\\ \mathbf{if}\;z \leq -1.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \mathsf{fma}\left(t, z, t\right)\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ t y) z))))
   (if (<= z -1.7)
     t_1
     (if (<= z -1.15e-199)
       (* x (- (/ y z) (fma t z t)))
       (if (<= z 1.0) (/ (* x (- y (* t z))) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((t + y) / z);
	double tmp;
	if (z <= -1.7) {
		tmp = t_1;
	} else if (z <= -1.15e-199) {
		tmp = x * ((y / z) - fma(t, z, t));
	} else if (z <= 1.0) {
		tmp = (x * (y - (t * z))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(t + y) / z))
	tmp = 0.0
	if (z <= -1.7)
		tmp = t_1;
	elseif (z <= -1.15e-199)
		tmp = Float64(x * Float64(Float64(y / z) - fma(t, z, t)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(x * Float64(y - Float64(t * z))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7], t$95$1, If[LessEqual[z, -1.15e-199], N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(x * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999996 or 1 < z

   1. Initial program 95.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 90.6

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 94.3

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

   11. Applied rewrites 94.3%

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

   if -1.69999999999999996 < z < -1.1500000000000001e-199

   1. Initial program 99.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -1.1500000000000001e-199 < z < 1

   1. Initial program 86.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. fp-cancel-sub-sign N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. fp-cancel-sub-sign N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

Alternative 4: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(-t\right)\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{+71}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- t))) (t_2 (* x (/ t z))))
   (if (<= t -7.2e+188)
     t_1
     (if (<= t -1.6e+71)
       t_2
       (if (<= t 1.8e+86) (* y (/ x z)) (if (<= t 4.2e+213) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * -t;
	double t_2 = x * (t / z);
	double tmp;
	if (t <= -7.2e+188) {
		tmp = t_1;
	} else if (t <= -1.6e+71) {
		tmp = t_2;
	} else if (t <= 1.8e+86) {
		tmp = y * (x / z);
	} else if (t <= 4.2e+213) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * -t
    t_2 = x * (t / z)
    if (t <= (-7.2d+188)) then
        tmp = t_1
    else if (t <= (-1.6d+71)) then
        tmp = t_2
    else if (t <= 1.8d+86) then
        tmp = y * (x / z)
    else if (t <= 4.2d+213) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * -t;
	double t_2 = x * (t / z);
	double tmp;
	if (t <= -7.2e+188) {
		tmp = t_1;
	} else if (t <= -1.6e+71) {
		tmp = t_2;
	} else if (t <= 1.8e+86) {
		tmp = y * (x / z);
	} else if (t <= 4.2e+213) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * -t
	t_2 = x * (t / z)
	tmp = 0
	if t <= -7.2e+188:
		tmp = t_1
	elif t <= -1.6e+71:
		tmp = t_2
	elif t <= 1.8e+86:
		tmp = y * (x / z)
	elif t <= 4.2e+213:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(-t))
	t_2 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t <= -7.2e+188)
		tmp = t_1;
	elseif (t <= -1.6e+71)
		tmp = t_2;
	elseif (t <= 1.8e+86)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 4.2e+213)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * -t;
	t_2 = x * (t / z);
	tmp = 0.0;
	if (t <= -7.2e+188)
		tmp = t_1;
	elseif (t <= -1.6e+71)
		tmp = t_2;
	elseif (t <= 1.8e+86)
		tmp = y * (x / z);
	elseif (t <= 4.2e+213)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+188], t$95$1, If[LessEqual[t, -1.6e+71], t$95$2, If[LessEqual[t, 1.8e+86], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+213], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.20000000000000041e188 or 1.80000000000000003e86 < t < 4.2000000000000001e213

   1. Initial program 96.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 59.6%

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

   if -7.20000000000000041e188 < t < -1.60000000000000012e71 or 4.2000000000000001e213 < t

   1. Initial program 94.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 68.1

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

   11. Applied rewrites 68.1%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 61.4%

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

   if -1.60000000000000012e71 < t < 1.80000000000000003e86

   1. Initial program 91.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 78.3%

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

Alternative 5: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+245}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-71}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 3700000000:\\ \;\;\;\;\frac{t \cdot x}{-1 + z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y x) z)))
   (if (<= y -1.3e+245)
     t_1
     (if (<= y -4.6e-71)
       (* (+ y t) (/ x z))
       (if (<= y 3700000000.0) (/ (* t x) (+ -1.0 z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * x) / z;
	double tmp;
	if (y <= -1.3e+245) {
		tmp = t_1;
	} else if (y <= -4.6e-71) {
		tmp = (y + t) * (x / z);
	} else if (y <= 3700000000.0) {
		tmp = (t * x) / (-1.0 + 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)
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) :: t_1
    real(8) :: tmp
    t_1 = (y * x) / z
    if (y <= (-1.3d+245)) then
        tmp = t_1
    else if (y <= (-4.6d-71)) then
        tmp = (y + t) * (x / z)
    else if (y <= 3700000000.0d0) then
        tmp = (t * x) / ((-1.0d0) + 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 t_1 = (y * x) / z;
	double tmp;
	if (y <= -1.3e+245) {
		tmp = t_1;
	} else if (y <= -4.6e-71) {
		tmp = (y + t) * (x / z);
	} else if (y <= 3700000000.0) {
		tmp = (t * x) / (-1.0 + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * x) / z
	tmp = 0
	if y <= -1.3e+245:
		tmp = t_1
	elif y <= -4.6e-71:
		tmp = (y + t) * (x / z)
	elif y <= 3700000000.0:
		tmp = (t * x) / (-1.0 + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (y <= -1.3e+245)
		tmp = t_1;
	elseif (y <= -4.6e-71)
		tmp = Float64(Float64(y + t) * Float64(x / z));
	elseif (y <= 3700000000.0)
		tmp = Float64(Float64(t * x) / Float64(-1.0 + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * x) / z;
	tmp = 0.0;
	if (y <= -1.3e+245)
		tmp = t_1;
	elseif (y <= -4.6e-71)
		tmp = (y + t) * (x / z);
	elseif (y <= 3700000000.0)
		tmp = (t * x) / (-1.0 + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.3e+245], t$95$1, If[LessEqual[y, -4.6e-71], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3700000000.0], N[(N[(t * x), $MachinePrecision] / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000002e245 or 3.7e9 < y

   1. Initial program 89.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 90.5%

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

   if -1.30000000000000002e245 < y < -4.5999999999999997e-71

   1. Initial program 93.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.0%

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

   if -4.5999999999999997e-71 < y < 3.7e9

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 6: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+245}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 3700000000:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y x) z)))
   (if (<= y -1.3e+245)
     t_1
     (if (<= y -6.5e-71)
       (* (+ y t) (/ x z))
       (if (<= y 3700000000.0) (* (/ x (- z 1.0)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * x) / z;
	double tmp;
	if (y <= -1.3e+245) {
		tmp = t_1;
	} else if (y <= -6.5e-71) {
		tmp = (y + t) * (x / z);
	} else if (y <= 3700000000.0) {
		tmp = (x / (z - 1.0)) * 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)
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) :: t_1
    real(8) :: tmp
    t_1 = (y * x) / z
    if (y <= (-1.3d+245)) then
        tmp = t_1
    else if (y <= (-6.5d-71)) then
        tmp = (y + t) * (x / z)
    else if (y <= 3700000000.0d0) then
        tmp = (x / (z - 1.0d0)) * 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 t_1 = (y * x) / z;
	double tmp;
	if (y <= -1.3e+245) {
		tmp = t_1;
	} else if (y <= -6.5e-71) {
		tmp = (y + t) * (x / z);
	} else if (y <= 3700000000.0) {
		tmp = (x / (z - 1.0)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * x) / z
	tmp = 0
	if y <= -1.3e+245:
		tmp = t_1
	elif y <= -6.5e-71:
		tmp = (y + t) * (x / z)
	elif y <= 3700000000.0:
		tmp = (x / (z - 1.0)) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * x) / z)
	tmp = 0.0
	if (y <= -1.3e+245)
		tmp = t_1;
	elseif (y <= -6.5e-71)
		tmp = Float64(Float64(y + t) * Float64(x / z));
	elseif (y <= 3700000000.0)
		tmp = Float64(Float64(x / Float64(z - 1.0)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * x) / z;
	tmp = 0.0;
	if (y <= -1.3e+245)
		tmp = t_1;
	elseif (y <= -6.5e-71)
		tmp = (y + t) * (x / z);
	elseif (y <= 3700000000.0)
		tmp = (x / (z - 1.0)) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.3e+245], t$95$1, If[LessEqual[y, -6.5e-71], N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3700000000.0], N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000002e245 or 3.7e9 < y

   1. Initial program 89.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 90.5%

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

   if -1.30000000000000002e245 < y < -6.50000000000000005e-71

   1. Initial program 93.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.0%

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

   if -6.50000000000000005e-71 < y < 3.7e9

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. associate-/l* N/A

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

      15. *-rgt-identity N/A

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

      16. associate-/r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

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

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

      21. *-commutative N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.9%

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

Alternative 7: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7) (not (<= z 1.0)))
   (* x (/ (+ t y) z))
   (/ (* x (- y (* t z))) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 1.0)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = (x * (y - (t * z))) / z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: tmp
    if ((z <= (-1.7d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * ((t + y) / z)
    else
        tmp = (x * (y - (t * z))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 1.0)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = (x * (y - (t * z))) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7) or not (z <= 1.0):
		tmp = x * ((t + y) / z)
	else:
		tmp = (x * (y - (t * z))) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7) || !(z <= 1.0))
		tmp = Float64(x * Float64(Float64(t + y) / z));
	else
		tmp = Float64(Float64(x * Float64(y - Float64(t * z))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.7) || ~((z <= 1.0)))
		tmp = x * ((t + y) / z);
	else
		tmp = (x * (y - (t * z))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999996 or 1 < z

   1. Initial program 95.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 90.6

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 94.3

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

   11. Applied rewrites 94.3%

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

   if -1.69999999999999996 < z < 1

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. fp-cancel-sub-sign N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. fp-cancel-sub-sign N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(-t\right)\\ \mathbf{if}\;t \leq -7.1 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- t))))
   (if (<= t -7.1e+71)
     t_1
     (if (<= t 1.8e+86)
       (* y (/ x z))
       (if (<= t 7.5e+224) t_1 (/ (* t x) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * -t;
	double tmp;
	if (t <= -7.1e+71) {
		tmp = t_1;
	} else if (t <= 1.8e+86) {
		tmp = y * (x / z);
	} else if (t <= 7.5e+224) {
		tmp = t_1;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_1
    real(8) :: tmp
    t_1 = x * -t
    if (t <= (-7.1d+71)) then
        tmp = t_1
    else if (t <= 1.8d+86) then
        tmp = y * (x / z)
    else if (t <= 7.5d+224) then
        tmp = t_1
    else
        tmp = (t * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * -t;
	double tmp;
	if (t <= -7.1e+71) {
		tmp = t_1;
	} else if (t <= 1.8e+86) {
		tmp = y * (x / z);
	} else if (t <= 7.5e+224) {
		tmp = t_1;
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * -t
	tmp = 0
	if t <= -7.1e+71:
		tmp = t_1
	elif t <= 1.8e+86:
		tmp = y * (x / z)
	elif t <= 7.5e+224:
		tmp = t_1
	else:
		tmp = (t * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(-t))
	tmp = 0.0
	if (t <= -7.1e+71)
		tmp = t_1;
	elseif (t <= 1.8e+86)
		tmp = Float64(y * Float64(x / z));
	elseif (t <= 7.5e+224)
		tmp = t_1;
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * -t;
	tmp = 0.0;
	if (t <= -7.1e+71)
		tmp = t_1;
	elseif (t <= 1.8e+86)
		tmp = y * (x / z);
	elseif (t <= 7.5e+224)
		tmp = t_1;
	else
		tmp = (t * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * (-t)), $MachinePrecision]}, If[LessEqual[t, -7.1e+71], t$95$1, If[LessEqual[t, 1.8e+86], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e+224], t$95$1, N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.09999999999999986e71 or 1.80000000000000003e86 < t < 7.500000000000001e224

   1. Initial program 96.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.4%

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

   if -7.09999999999999986e71 < t < 1.80000000000000003e86

   1. Initial program 91.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 77.9%

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

   if 7.500000000000001e224 < t

   1. Initial program 90.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.3%

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

Alternative 9: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7) (not (<= z 9e-9)))
   (* x (/ (+ t y) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 9e-9)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	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)
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) :: tmp
    if ((z <= (-1.7d0)) .or. (.not. (z <= 9d-9))) then
        tmp = x * ((t + y) / z)
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 9e-9)) {
		tmp = x * ((t + y) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7) or not (z <= 9e-9):
		tmp = x * ((t + y) / z)
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7) || !(z <= 9e-9))
		tmp = Float64(x * Float64(Float64(t + y) / z));
	else
		tmp = Float64(x * Float64(Float64(y / z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.7) || ~((z <= 9e-9)))
		tmp = x * ((t + y) / z);
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7], N[Not[LessEqual[z, 9e-9]], $MachinePrecision]], N[(x * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999996 or 8.99999999999999953e-9 < z

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 90.7

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in z around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 94.3

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

   11. Applied rewrites 94.3%

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

   if -1.69999999999999996 < z < 8.99999999999999953e-9

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7) (not (<= z 9e-9)))
   (/ (* (+ t y) x) z)
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 9e-9)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = x * ((y / z) - t);
	}
	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)
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) :: tmp
    if ((z <= (-1.7d0)) .or. (.not. (z <= 9d-9))) then
        tmp = ((t + y) * x) / z
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7) || !(z <= 9e-9)) {
		tmp = ((t + y) * x) / z;
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7) or not (z <= 9e-9):
		tmp = ((t + y) * x) / z
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7) || !(z <= 9e-9))
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	else
		tmp = Float64(x * Float64(Float64(y / z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.7) || ~((z <= 9e-9)))
		tmp = ((t + y) * x) / z;
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7], N[Not[LessEqual[z, 9e-9]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999996 or 8.99999999999999953e-9 < z

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -1.69999999999999996 < z < 8.99999999999999953e-9

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \lor \neg \left(z \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-79} \lor \neg \left(y \leq 3700000000\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.7e-79) (not (<= y 3700000000.0)))
   (/ (* y x) z)
   (* (/ x (- z 1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.7e-79) || !(y <= 3700000000.0)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / (z - 1.0)) * t;
	}
	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)
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) :: tmp
    if ((y <= (-2.7d-79)) .or. (.not. (y <= 3700000000.0d0))) then
        tmp = (y * x) / z
    else
        tmp = (x / (z - 1.0d0)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.7e-79) || !(y <= 3700000000.0)) {
		tmp = (y * x) / z;
	} else {
		tmp = (x / (z - 1.0)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.7e-79) or not (y <= 3700000000.0):
		tmp = (y * x) / z
	else:
		tmp = (x / (z - 1.0)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.7e-79) || !(y <= 3700000000.0))
		tmp = Float64(Float64(y * x) / z);
	else
		tmp = Float64(Float64(x / Float64(z - 1.0)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.7e-79) || ~((y <= 3700000000.0)))
		tmp = (y * x) / z;
	else
		tmp = (x / (z - 1.0)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.7e-79], N[Not[LessEqual[y, 3700000000.0]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7000000000000002e-79 or 3.7e9 < y

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 79.4%

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

   if -2.7000000000000002e-79 < y < 3.7e9

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. associate-/l* N/A

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

      15. *-rgt-identity N/A

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

      16. associate-/r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

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

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

      21. *-commutative N/A

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

      22. mul-1-neg N/A

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

      23. remove-double-neg N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.4%

      \[\leadsto \frac{x}{z - 1} \cdot \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-79} \lor \neg \left(y \leq 3700000000\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - 1} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \mathbf{elif}\;y \leq 3700000000:\\ \;\;\;\;\frac{t \cdot x}{-1 + z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.5e-71)
   (/ (* (+ t y) x) z)
   (if (<= y 3700000000.0) (/ (* t x) (+ -1.0 z)) (/ (* y x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e-71) {
		tmp = ((t + y) * x) / z;
	} else if (y <= 3700000000.0) {
		tmp = (t * x) / (-1.0 + z);
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: tmp
    if (y <= (-6.5d-71)) then
        tmp = ((t + y) * x) / z
    else if (y <= 3700000000.0d0) then
        tmp = (t * x) / ((-1.0d0) + z)
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e-71) {
		tmp = ((t + y) * x) / z;
	} else if (y <= 3700000000.0) {
		tmp = (t * x) / (-1.0 + z);
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.5e-71:
		tmp = ((t + y) * x) / z
	elif y <= 3700000000.0:
		tmp = (t * x) / (-1.0 + z)
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.5e-71)
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	elseif (y <= 3700000000.0)
		tmp = Float64(Float64(t * x) / Float64(-1.0 + z));
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -6.5e-71)
		tmp = ((t + y) * x) / z;
	elseif (y <= 3700000000.0)
		tmp = (t * x) / (-1.0 + z);
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.5e-71], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 3700000000.0], N[(N[(t * x), $MachinePrecision] / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.50000000000000005e-71

   1. Initial program 90.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -6.50000000000000005e-71 < y < 3.7e9

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if 3.7e9 < y

   1. Initial program 91.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 88.5%

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

Alternative 13: 41.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.75) (not (<= z 1.0))) (/ (* t x) z) (* (- t) (fma z x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.75) || !(z <= 1.0)) {
		tmp = (t * x) / z;
	} else {
		tmp = -t * fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.75) || !(z <= 1.0))
		tmp = Float64(Float64(t * x) / z);
	else
		tmp = Float64(Float64(-t) * fma(z, x, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.75], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision], N[((-t) * N[(z * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1 < z

   1. Initial program 95.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.3%

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

   if -0.75 < z < 1

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 41.4

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

   5. Applied rewrites 41.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 40.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.75 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.75)
   (* (/ x z) t)
   (if (<= z 1.0) (* (- t) (fma z x x)) (/ (* t x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.75) {
		tmp = (x / z) * t;
	} else if (z <= 1.0) {
		tmp = -t * fma(z, x, x);
	} else {
		tmp = (t * x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.75)
		tmp = Float64(Float64(x / z) * t);
	elseif (z <= 1.0)
		tmp = Float64(Float64(-t) * fma(z, x, x));
	else
		tmp = Float64(Float64(t * x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.75], N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 1.0], N[((-t) * N[(z * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.75

   1. Initial program 95.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 48.4%

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

   if -0.75 < z < 1

   1. Initial program 91.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 41.4

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

   5. Applied rewrites 41.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 40.7%

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

   if 1 < z

   1. Initial program 96.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 59.6%

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

Alternative 15: 22.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * (-t)), $MachinePrecision]

Derivation

1. Initial program 93.1%

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

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

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

   4. div-sub N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 66.8

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 29.9%

   \[\leadsto x \cdot \left(-t\right) \]
9. Add Preprocessing

Developer Target 1: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - 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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - 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 t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024363 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- (/ y z) (/ t (- 1 z)))) -3811613151656021/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 7066972463851151/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z))))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))