Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.5% → 96.8%

Time: 4.0s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 9.6e-57)
    (/ (* x_m (- y z)) (- t z))
    (* (/ x_m (- t z)) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 9.6e-57) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 9.6d-57) then
        tmp = (x_m * (y - z)) / (t - z)
    else
        tmp = (x_m / (t - z)) * (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 9.6e-57) {
		tmp = (x_m * (y - z)) / (t - z);
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 9.6e-57:
		tmp = (x_m * (y - z)) / (t - z)
	else:
		tmp = (x_m / (t - z)) * (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 9.6e-57)
		tmp = Float64(Float64(x_m * Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 9.6e-57)
		tmp = (x_m * (y - z)) / (t - z);
	else
		tmp = (x_m / (t - z)) * (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 9.6e-57], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.60000000000000025e-57

   1. Initial program 84.5%

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

   if 9.60000000000000025e-57 < x

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.5

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

   3. Applied rewrites 84.5%

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

Alternative 2: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{z - y}{z - t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.05e+35)
    (* (/ (- z y) (- z t)) x_m)
    (* (/ x_m (- t z)) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.05e+35) {
		tmp = ((z - y) / (z - t)) * x_m;
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 2.05d+35) then
        tmp = ((z - y) / (z - t)) * x_m
    else
        tmp = (x_m / (t - z)) * (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 2.05e+35) {
		tmp = ((z - y) / (z - t)) * x_m;
	} else {
		tmp = (x_m / (t - z)) * (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 2.05e+35:
		tmp = ((z - y) / (z - t)) * x_m
	else:
		tmp = (x_m / (t - z)) * (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 2.05e+35)
		tmp = Float64(Float64(Float64(z - y) / Float64(z - t)) * x_m);
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 2.05e+35)
		tmp = ((z - y) / (z - t)) * x_m;
	else
		tmp = (x_m / (t - z)) * (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 2.05e+35], N[(N[(N[(z - y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999999e35

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   if 2.0499999999999999e35 < x

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.5

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

   3. Applied rewrites 84.5%

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

Alternative 3: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.46 \cdot 10^{+88}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.46e+88)
    (/ x_m (/ z (- z y)))
    (if (<= z 2.35e+170) (* (/ x_m (- t z)) (- y z)) (* (/ (- z y) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.46e+88) {
		tmp = x_m / (z / (z - y));
	} else if (z <= 2.35e+170) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = ((z - y) / z) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.46d+88)) then
        tmp = x_m / (z / (z - y))
    else if (z <= 2.35d+170) then
        tmp = (x_m / (t - z)) * (y - z)
    else
        tmp = ((z - y) / z) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.46e+88) {
		tmp = x_m / (z / (z - y));
	} else if (z <= 2.35e+170) {
		tmp = (x_m / (t - z)) * (y - z);
	} else {
		tmp = ((z - y) / z) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.46e+88:
		tmp = x_m / (z / (z - y))
	elif z <= 2.35e+170:
		tmp = (x_m / (t - z)) * (y - z)
	else:
		tmp = ((z - y) / z) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.46e+88)
		tmp = Float64(x_m / Float64(z / Float64(z - y)));
	elseif (z <= 2.35e+170)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(y - z));
	else
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.46e+88)
		tmp = x_m / (z / (z - y));
	elseif (z <= 2.35e+170)
		tmp = (x_m / (t - z)) * (y - z);
	else
		tmp = ((z - y) / z) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.46e+88], N[(x$95$m / N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.35e+170], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.45999999999999999e88

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.9

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

   6. Applied rewrites 52.9%

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

   if -1.45999999999999999e88 < z < 2.35000000000000002e170

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.5

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

   3. Applied rewrites 84.5%

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

   if 2.35000000000000002e170 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

Alternative 4: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{z - y}}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x\_m \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (/ x_m (/ z (- z y)))
    (if (<= z 2.1e-11) (/ (* x_m y) (- t z)) (* (/ z (- z t)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = x_m / (z / (z - y));
	} else if (z <= 2.1e-11) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = x_m / (z / (z - y))
    else if (z <= 2.1d-11) then
        tmp = (x_m * y) / (t - z)
    else
        tmp = (z / (z - t)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = x_m / (z / (z - y));
	} else if (z <= 2.1e-11) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = x_m / (z / (z - y))
	elif z <= 2.1e-11:
		tmp = (x_m * y) / (t - z)
	else:
		tmp = (z / (z - t)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(x_m / Float64(z / Float64(z - y)));
	elseif (z <= 2.1e-11)
		tmp = Float64(Float64(x_m * y) / Float64(t - z));
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = x_m / (z / (z - y));
	elseif (z <= 2.1e-11)
		tmp = (x_m * y) / (t - z);
	else
		tmp = (z / (z - t)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(x$95$m / N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-11], N[(N[(x$95$m * y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e-7

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.9

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

   6. Applied rewrites 52.9%

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

   if -3.2000000000000001e-7 < z < 2.0999999999999999e-11

   1. Initial program 84.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if 2.0999999999999999e-11 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 54.0%

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

Alternative 5: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x\_m \cdot y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (* (/ (- z y) z) x_m)
    (if (<= z 2.1e-11) (/ (* x_m y) (- t z)) (* (/ z (- z t)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= 2.1e-11) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = ((z - y) / z) * x_m
    else if (z <= 2.1d-11) then
        tmp = (x_m * y) / (t - z)
    else
        tmp = (z / (z - t)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= 2.1e-11) {
		tmp = (x_m * y) / (t - z);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = ((z - y) / z) * x_m
	elif z <= 2.1e-11:
		tmp = (x_m * y) / (t - z)
	else:
		tmp = (z / (z - t)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	elseif (z <= 2.1e-11)
		tmp = Float64(Float64(x_m * y) / Float64(t - z));
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = ((z - y) / z) * x_m;
	elseif (z <= 2.1e-11)
		tmp = (x_m * y) / (t - z);
	else
		tmp = (z / (z - t)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 2.1e-11], N[(N[(x$95$m * y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e-7

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   if -3.2000000000000001e-7 < z < 2.0999999999999999e-11

   1. Initial program 84.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if 2.0999999999999999e-11 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 54.0%

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

Alternative 6: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-183}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -7.2e-8)
    (* (/ (- z y) z) x_m)
    (if (<= z -1.05e-183)
      (/ (* x_m (- y z)) t)
      (if (<= z 4.7e-18) (* (- y z) (/ x_m t)) (* (/ z (- z t)) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -7.2e-8) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= -1.05e-183) {
		tmp = (x_m * (y - z)) / t;
	} else if (z <= 4.7e-18) {
		tmp = (y - z) * (x_m / t);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7.2d-8)) then
        tmp = ((z - y) / z) * x_m
    else if (z <= (-1.05d-183)) then
        tmp = (x_m * (y - z)) / t
    else if (z <= 4.7d-18) then
        tmp = (y - z) * (x_m / t)
    else
        tmp = (z / (z - t)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -7.2e-8) {
		tmp = ((z - y) / z) * x_m;
	} else if (z <= -1.05e-183) {
		tmp = (x_m * (y - z)) / t;
	} else if (z <= 4.7e-18) {
		tmp = (y - z) * (x_m / t);
	} else {
		tmp = (z / (z - t)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -7.2e-8:
		tmp = ((z - y) / z) * x_m
	elif z <= -1.05e-183:
		tmp = (x_m * (y - z)) / t
	elif z <= 4.7e-18:
		tmp = (y - z) * (x_m / t)
	else:
		tmp = (z / (z - t)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -7.2e-8)
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	elseif (z <= -1.05e-183)
		tmp = Float64(Float64(x_m * Float64(y - z)) / t);
	elseif (z <= 4.7e-18)
		tmp = Float64(Float64(y - z) * Float64(x_m / t));
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -7.2e-8)
		tmp = ((z - y) / z) * x_m;
	elseif (z <= -1.05e-183)
		tmp = (x_m * (y - z)) / t;
	elseif (z <= 4.7e-18)
		tmp = (y - z) * (x_m / t);
	else
		tmp = (z / (z - t)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -7.2e-8], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, -1.05e-183], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 4.7e-18], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -7.19999999999999962e-8

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   if -7.19999999999999962e-8 < z < -1.0500000000000001e-183

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -1.0500000000000001e-183 < z < 4.6999999999999996e-18

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.6

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

   6. Applied rewrites 47.6%

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

   if 4.6999999999999996e-18 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 54.0%

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

Alternative 7: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -105000000000:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq 0.00055:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -105000000000.0)
    (* (- y z) (/ x_m t))
    (if (<= t 0.00055) (* (/ (- z y) z) x_m) (/ (* x_m (- y z)) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -105000000000.0) {
		tmp = (y - z) * (x_m / t);
	} else if (t <= 0.00055) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = (x_m * (y - z)) / t;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-105000000000.0d0)) then
        tmp = (y - z) * (x_m / t)
    else if (t <= 0.00055d0) then
        tmp = ((z - y) / z) * x_m
    else
        tmp = (x_m * (y - z)) / t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -105000000000.0) {
		tmp = (y - z) * (x_m / t);
	} else if (t <= 0.00055) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = (x_m * (y - z)) / t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -105000000000.0:
		tmp = (y - z) * (x_m / t)
	elif t <= 0.00055:
		tmp = ((z - y) / z) * x_m
	else:
		tmp = (x_m * (y - z)) / t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -105000000000.0)
		tmp = Float64(Float64(y - z) * Float64(x_m / t));
	elseif (t <= 0.00055)
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	else
		tmp = Float64(Float64(x_m * Float64(y - z)) / t);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -105000000000.0)
		tmp = (y - z) * (x_m / t);
	elseif (t <= 0.00055)
		tmp = ((z - y) / z) * x_m;
	else
		tmp = (x_m * (y - z)) / t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -105000000000.0], N[(N[(y - z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.00055], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e11

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.6

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

   6. Applied rewrites 47.6%

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

   if -1.05e11 < t < 5.50000000000000033e-4

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   if 5.50000000000000033e-4 < t

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.4

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

   9. Applied rewrites 47.4%

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

Alternative 8: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -105000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.00055:\\ \;\;\;\;\frac{z - y}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) t)))
   (*
    x_s
    (if (<= t -105000000000.0)
      t_1
      (if (<= t 0.00055) (* (/ (- z y) z) x_m) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / t;
	double tmp;
	if (t <= -105000000000.0) {
		tmp = t_1;
	} else if (t <= 0.00055) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m * (y - z)) / t
    if (t <= (-105000000000.0d0)) then
        tmp = t_1
    else if (t <= 0.00055d0) then
        tmp = ((z - y) / z) * x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / t;
	double tmp;
	if (t <= -105000000000.0) {
		tmp = t_1;
	} else if (t <= 0.00055) {
		tmp = ((z - y) / z) * x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * (y - z)) / t
	tmp = 0
	if t <= -105000000000.0:
		tmp = t_1
	elif t <= 0.00055:
		tmp = ((z - y) / z) * x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / t)
	tmp = 0.0
	if (t <= -105000000000.0)
		tmp = t_1;
	elseif (t <= 0.00055)
		tmp = Float64(Float64(Float64(z - y) / z) * x_m);
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * (y - z)) / t;
	tmp = 0.0;
	if (t <= -105000000000.0)
		tmp = t_1;
	elseif (t <= 0.00055)
		tmp = ((z - y) / z) * x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -105000000000.0], t$95$1, If[LessEqual[t, 0.00055], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e11 or 5.50000000000000033e-4 < t

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -1.05e11 < t < 5.50000000000000033e-4

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

Alternative 9: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot \left(y - z\right)}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -105000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (* x_m (- y z)) t)))
   (*
    x_s
    (if (<= t -105000000000.0)
      t_1
      (if (<= t 1.1e-8) (* (/ x_m z) (- z y)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / t;
	double tmp;
	if (t <= -105000000000.0) {
		tmp = t_1;
	} else if (t <= 1.1e-8) {
		tmp = (x_m / z) * (z - y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m * (y - z)) / t
    if (t <= (-105000000000.0d0)) then
        tmp = t_1
    else if (t <= 1.1d-8) then
        tmp = (x_m / z) * (z - y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m * (y - z)) / t;
	double tmp;
	if (t <= -105000000000.0) {
		tmp = t_1;
	} else if (t <= 1.1e-8) {
		tmp = (x_m / z) * (z - y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m * (y - z)) / t
	tmp = 0
	if t <= -105000000000.0:
		tmp = t_1
	elif t <= 1.1e-8:
		tmp = (x_m / z) * (z - y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / t)
	tmp = 0.0
	if (t <= -105000000000.0)
		tmp = t_1;
	elseif (t <= 1.1e-8)
		tmp = Float64(Float64(x_m / z) * Float64(z - y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m * (y - z)) / t;
	tmp = 0.0;
	if (t <= -105000000000.0)
		tmp = t_1;
	elseif (t <= 1.1e-8)
		tmp = (x_m / z) * (z - y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -105000000000.0], t$95$1, If[LessEqual[t, 1.1e-8], N[(N[(x$95$m / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e11 or 1.0999999999999999e-8 < t

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if -1.05e11 < t < 1.0999999999999999e-8

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 43.7

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

   8. Applied rewrites 43.7%

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

Alternative 10: 65.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{+41}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{x\_m \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.48e+41)
    (* 1.0 x_m)
    (if (<= z 5.1e-18)
      (/ (* x_m (- y z)) t)
      (if (<= z 2.7e+170) (/ (* x_m z) (- z t)) (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.48e+41) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.1e-18) {
		tmp = (x_m * (y - z)) / t;
	} else if (z <= 2.7e+170) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.48d+41)) then
        tmp = 1.0d0 * x_m
    else if (z <= 5.1d-18) then
        tmp = (x_m * (y - z)) / t
    else if (z <= 2.7d+170) then
        tmp = (x_m * z) / (z - t)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.48e+41) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.1e-18) {
		tmp = (x_m * (y - z)) / t;
	} else if (z <= 2.7e+170) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.48e+41:
		tmp = 1.0 * x_m
	elif z <= 5.1e-18:
		tmp = (x_m * (y - z)) / t
	elif z <= 2.7e+170:
		tmp = (x_m * z) / (z - t)
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.48e+41)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 5.1e-18)
		tmp = Float64(Float64(x_m * Float64(y - z)) / t);
	elseif (z <= 2.7e+170)
		tmp = Float64(Float64(x_m * z) / Float64(z - t));
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.48e+41)
		tmp = 1.0 * x_m;
	elseif (z <= 5.1e-18)
		tmp = (x_m * (y - z)) / t;
	elseif (z <= 2.7e+170)
		tmp = (x_m * z) / (z - t);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.48e+41], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 5.1e-18], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.7e+170], N[(N[(x$95$m * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.48e41 or 2.7000000000000002e170 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -1.48e41 < z < 5.09999999999999983e-18

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 37.4

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.4

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

   9. Applied rewrites 47.4%

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

   if 5.09999999999999983e-18 < z < 2.7000000000000002e170

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.9

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

   6. Applied rewrites 44.9%

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

Alternative 11: 63.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{x\_m \cdot z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (* 1.0 x_m)
    (if (<= z 2.6e-18)
      (/ x_m (/ t y))
      (if (<= z 2.7e+170) (/ (* x_m z) (- z t)) (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 2.6e-18) {
		tmp = x_m / (t / y);
	} else if (z <= 2.7e+170) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = 1.0d0 * x_m
    else if (z <= 2.6d-18) then
        tmp = x_m / (t / y)
    else if (z <= 2.7d+170) then
        tmp = (x_m * z) / (z - t)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 2.6e-18) {
		tmp = x_m / (t / y);
	} else if (z <= 2.7e+170) {
		tmp = (x_m * z) / (z - t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = 1.0 * x_m
	elif z <= 2.6e-18:
		tmp = x_m / (t / y)
	elif z <= 2.7e+170:
		tmp = (x_m * z) / (z - t)
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 2.6e-18)
		tmp = Float64(x_m / Float64(t / y));
	elseif (z <= 2.7e+170)
		tmp = Float64(Float64(x_m * z) / Float64(z - t));
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = 1.0 * x_m;
	elseif (z <= 2.6e-18)
		tmp = x_m / (t / y);
	elseif (z <= 2.7e+170)
		tmp = (x_m * z) / (z - t);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 2.6e-18], N[(x$95$m / N[(t / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+170], N[(N[(x$95$m * z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e-7 or 2.7000000000000002e170 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -3.2000000000000001e-7 < z < 2.6e-18

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 39.4

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

   6. Applied rewrites 39.4%

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

   if 2.6e-18 < z < 2.7000000000000002e170

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 44.9

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

   6. Applied rewrites 44.9%

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

Alternative 12: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-185}:\\ \;\;\;\;\frac{x\_m \cdot y}{t}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (* 1.0 x_m)
    (if (<= z -4e-185)
      (/ (* x_m y) t)
      (if (<= z 4.7e-18) (* y (/ x_m t)) (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= -4e-185) {
		tmp = (x_m * y) / t;
	} else if (z <= 4.7e-18) {
		tmp = y * (x_m / t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = 1.0d0 * x_m
    else if (z <= (-4d-185)) then
        tmp = (x_m * y) / t
    else if (z <= 4.7d-18) then
        tmp = y * (x_m / t)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= -4e-185) {
		tmp = (x_m * y) / t;
	} else if (z <= 4.7e-18) {
		tmp = y * (x_m / t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = 1.0 * x_m
	elif z <= -4e-185:
		tmp = (x_m * y) / t
	elif z <= 4.7e-18:
		tmp = y * (x_m / t)
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(1.0 * x_m);
	elseif (z <= -4e-185)
		tmp = Float64(Float64(x_m * y) / t);
	elseif (z <= 4.7e-18)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = 1.0 * x_m;
	elseif (z <= -4e-185)
		tmp = (x_m * y) / t;
	elseif (z <= 4.7e-18)
		tmp = y * (x_m / t);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, -4e-185], N[(N[(x$95$m * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 4.7e-18], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000001e-7 or 4.6999999999999996e-18 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -3.2000000000000001e-7 < z < -4e-185

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

   if -4e-185 < z < 4.6999999999999996e-18

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{t} \]

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.6

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

   6. Applied rewrites 37.6%

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

Alternative 13: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (* 1.0 x_m)
    (if (<= z 5.5e-18) (/ x_m (/ t y)) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.5e-18) {
		tmp = x_m / (t / y);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = 1.0d0 * x_m
    else if (z <= 5.5d-18) then
        tmp = x_m / (t / y)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.5e-18) {
		tmp = x_m / (t / y);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = 1.0 * x_m
	elif z <= 5.5e-18:
		tmp = x_m / (t / y)
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 5.5e-18)
		tmp = Float64(x_m / Float64(t / y));
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = 1.0 * x_m;
	elseif (z <= 5.5e-18)
		tmp = x_m / (t / y);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 5.5e-18], N[(x$95$m / N[(t / y), $MachinePrecision]), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e-7 or 5.5e-18 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -3.2000000000000001e-7 < z < 5.5e-18

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 39.4

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

   6. Applied rewrites 39.4%

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

Alternative 14: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.2e-7)
    (* 1.0 x_m)
    (if (<= z 5.5e-18) (* (/ y t) x_m) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.5e-18) {
		tmp = (y / t) * x_m;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.2d-7)) then
        tmp = 1.0d0 * x_m
    else if (z <= 5.5d-18) then
        tmp = (y / t) * x_m
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.2e-7) {
		tmp = 1.0 * x_m;
	} else if (z <= 5.5e-18) {
		tmp = (y / t) * x_m;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.2e-7:
		tmp = 1.0 * x_m
	elif z <= 5.5e-18:
		tmp = (y / t) * x_m
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.2e-7)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 5.5e-18)
		tmp = Float64(Float64(y / t) * x_m);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.2e-7)
		tmp = 1.0 * x_m;
	elseif (z <= 5.5e-18)
		tmp = (y / t) * x_m;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.2e-7], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 5.5e-18], N[(N[(y / t), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e-7 or 5.5e-18 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -3.2000000000000001e-7 < z < 5.5e-18

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 39.4

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

   6. Applied rewrites 39.4%

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

Alternative 15: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot x\_m\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.12e-8)
    (* 1.0 x_m)
    (if (<= z 4.7e-18) (* y (/ x_m t)) (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.12e-8) {
		tmp = 1.0 * x_m;
	} else if (z <= 4.7e-18) {
		tmp = y * (x_m / t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.12d-8)) then
        tmp = 1.0d0 * x_m
    else if (z <= 4.7d-18) then
        tmp = y * (x_m / t)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.12e-8) {
		tmp = 1.0 * x_m;
	} else if (z <= 4.7e-18) {
		tmp = y * (x_m / t);
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.12e-8:
		tmp = 1.0 * x_m
	elif z <= 4.7e-18:
		tmp = y * (x_m / t)
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.12e-8)
		tmp = Float64(1.0 * x_m);
	elseif (z <= 4.7e-18)
		tmp = Float64(y * Float64(x_m / t));
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.12e-8)
		tmp = 1.0 * x_m;
	elseif (z <= 4.7e-18)
		tmp = y * (x_m / t);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.12e-8], N[(1.0 * x$95$m), $MachinePrecision], If[LessEqual[z, 4.7e-18], N[(y * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.11999999999999994e-8 or 4.6999999999999996e-18 < z

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 97.0

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

   3. Applied rewrites 97.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.4%

      \[\leadsto 1 \cdot x \]

   if -1.11999999999999994e-8 < z < 4.6999999999999996e-18

   1. Initial program 84.5%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.5

         \[\leadsto \frac{x \cdot y}{t} \]

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{t} \]

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.6

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

   6. Applied rewrites 37.6%

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

Alternative 16: 35.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(1 \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* 1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (1.0 * x_m);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (1.0d0 * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (1.0 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (1.0 * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(1.0 * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (1.0 * x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

   8. distribute-frac-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f64 97.0

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

3. Applied rewrites 97.0%

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

4. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 52.8

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

6. Applied rewrites 52.8%

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

7. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
8. Step-by-step derivation

9. Applied rewrites 35.4%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64
  (/ (* x (- y z)) (- t z)))