Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.4% → 99.6%

Time: 5.1s

Alternatives: 8

Speedup: 0.8×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(1 + \left(y - z\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-52)
    (/ (fma (- y z) x_m x_m) z)
    (* (/ x_m z) (+ 1.0 (- 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 tmp;
	if (x_m <= 1e-52) {
		tmp = fma((y - z), x_m, x_m) / z;
	} else {
		tmp = (x_m / z) * (1.0 + (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)
	tmp = 0.0
	if (x_m <= 1e-52)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 + Float64(y - z)));
	end
	return Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-52], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e-52

   1. Initial program 93.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if 1e-52 < x

   1. Initial program 78.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+207} \lor \neg \left(z \leq 2.35 \cdot 10^{+150}\right):\\ \;\;\;\;-x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.15e+207) (not (<= z 2.35e+150)))
    (- x_m)
    (/ (fma (- y z) x_m x_m) 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 tmp;
	if ((z <= -2.15e+207) || !(z <= 2.35e+150)) {
		tmp = -x_m;
	} else {
		tmp = fma((y - z), x_m, x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -2.15e+207) || !(z <= 2.35e+150))
		tmp = Float64(-x_m);
	else
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	end
	return Float64(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_] := N[(x$95$s * If[Or[LessEqual[z, -2.15e+207], N[Not[LessEqual[z, 2.35e+150]], $MachinePrecision]], (-x$95$m), N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e207 or 2.35000000000000002e150 < z

   1. Initial program 64.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 92.0

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{-x} \]

   if -2.1499999999999999e207 < z < 2.35000000000000002e150

   1. Initial program 96.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 96.6

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

   4. Applied rewrites 96.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+207} \lor \neg \left(z \leq 2.35 \cdot 10^{+150}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -5e-5)
    (- (/ x_m z) x_m)
    (if (<= z 1.08e+70) (/ (fma y x_m x_m) 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 tmp;
	if (z <= -5e-5) {
		tmp = (x_m / z) - x_m;
	} else if (z <= 1.08e+70) {
		tmp = fma(y, x_m, x_m) / z;
	} else {
		tmp = -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)
	tmp = 0.0
	if (z <= -5e-5)
		tmp = Float64(Float64(x_m / z) - x_m);
	elseif (z <= 1.08e+70)
		tmp = Float64(fma(y, x_m, x_m) / z);
	else
		tmp = Float64(-x_m);
	end
	return Float64(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_] := N[(x$95$s * If[LessEqual[z, -5e-5], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], If[LessEqual[z, 1.08e+70], N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], (-x$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000024e-5

   1. Initial program 79.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. times-frac N/A

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

      3. /-rgt-identity N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -5.00000000000000024e-5 < z < 1.0799999999999999e70

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.0799999999999999e70 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 82.3

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 82.3%

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

Alternative 4: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+21} \lor \neg \left(y \leq 9\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -7.2e+21) (not (<= y 9.0)))
    (* (/ x_m z) y)
    (- (/ x_m 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 tmp;
	if ((y <= -7.2e+21) || !(y <= 9.0)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (x_m / 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)
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) :: tmp
    if ((y <= (-7.2d+21)) .or. (.not. (y <= 9.0d0))) then
        tmp = (x_m / z) * y
    else
        tmp = (x_m / 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 tmp;
	if ((y <= -7.2e+21) || !(y <= 9.0)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (x_m / 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):
	tmp = 0
	if (y <= -7.2e+21) or not (y <= 9.0):
		tmp = (x_m / z) * y
	else:
		tmp = (x_m / 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)
	tmp = 0.0
	if ((y <= -7.2e+21) || !(y <= 9.0))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(Float64(x_m / 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)
	tmp = 0.0;
	if ((y <= -7.2e+21) || ~((y <= 9.0)))
		tmp = (x_m / z) * y;
	else
		tmp = (x_m / 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_] := N[(x$95$s * If[Or[LessEqual[y, -7.2e+21], N[Not[LessEqual[y, 9.0]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e21 or 9 < y

   1. Initial program 87.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 91.7

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

   4. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 91.7

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

   6. Applied rewrites 91.7%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 76.7

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

   9. Applied rewrites 76.7%

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

   if -7.2e21 < y < 9

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. times-frac N/A

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

      3. /-rgt-identity N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+21} \lor \neg \left(y \leq 9\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{elif}\;y \leq 9:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -7.2e+21)
    (* (/ x_m z) y)
    (if (<= y 9.0) (- (/ x_m z) x_m) (/ (* y x_m) 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 tmp;
	if (y <= -7.2e+21) {
		tmp = (x_m / z) * y;
	} else if (y <= 9.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (y * x_m) / 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)
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) :: tmp
    if (y <= (-7.2d+21)) then
        tmp = (x_m / z) * y
    else if (y <= 9.0d0) then
        tmp = (x_m / z) - x_m
    else
        tmp = (y * x_m) / 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 tmp;
	if (y <= -7.2e+21) {
		tmp = (x_m / z) * y;
	} else if (y <= 9.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (y * x_m) / 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):
	tmp = 0
	if y <= -7.2e+21:
		tmp = (x_m / z) * y
	elif y <= 9.0:
		tmp = (x_m / z) - x_m
	else:
		tmp = (y * x_m) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -7.2e+21)
		tmp = Float64(Float64(x_m / z) * y);
	elseif (y <= 9.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(Float64(y * x_m) / 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)
	tmp = 0.0;
	if (y <= -7.2e+21)
		tmp = (x_m / z) * y;
	elseif (y <= 9.0)
		tmp = (x_m / z) - x_m;
	else
		tmp = (y * x_m) / 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_] := N[(x$95$s * If[LessEqual[y, -7.2e+21], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 9.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e21

   1. Initial program 85.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.2

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

   9. Applied rewrites 78.2%

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

   if -7.2e21 < y < 9

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. times-frac N/A

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

      3. /-rgt-identity N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. *-rgt-identity N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 9 < y

   1. Initial program 88.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 9:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 + y}{z} - 1\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.4e-75)
    (/ (fma (- y z) x_m x_m) z)
    (* (- (/ (+ 1.0 y) z) 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 tmp;
	if (x_m <= 1.4e-75) {
		tmp = fma((y - z), x_m, x_m) / z;
	} else {
		tmp = (((1.0 + y) / z) - 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)
	tmp = 0.0
	if (x_m <= 1.4e-75)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + y) / z) - 1.0) * x_m);
	end
	return Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.4e-75], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision] - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.39999999999999999e-75

   1. Initial program 93.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 93.1

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

   4. Applied rewrites 93.1%

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

   if 1.39999999999999999e-75 < x

   1. Initial program 80.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 99.9

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

      4. lift-/.f64 N/A

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

      5. *-inverses 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 7: 66.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{x\_m}{z} - x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- (/ x_m 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) {
	return x_s * ((x_m / z) - 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)
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
    code = x_s * ((x_m / z) - 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) {
	return x_s * ((x_m / z) - x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * ((x_m / z) - x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) - x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * ((x_m / z) - 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_] := N[(x$95$s * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. times-frac N/A

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

   3. /-rgt-identity N/A

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

   4. div-sub N/A

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

   5. *-inverses N/A

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

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

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

   7. *-rgt-identity N/A

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

   8. associate-*r/ N/A

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

   9. *-rgt-identity N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 64.2

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

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{\frac{x}{z} - x} \]
6. Add Preprocessing

Alternative 8: 38.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- 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) {
	return x_s * -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)
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
    code = x_s * -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) {
	return x_s * -x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * -x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(-x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * -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_] := N[(x$95$s * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 35.2

      \[\leadsto \color{blue}{-x} \]

5. Applied rewrites 35.2%

   \[\leadsto \color{blue}{-x} \]
6. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024354 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

  (/ (* x (+ (- y z) 1.0)) z))