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

Percentage Accurate: 88.1% → 99.8%

Time: 2.6s

Alternatives: 8

Speedup: 1.1×

• 21.0% 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: 88.1% 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 75000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z\right) - -1\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= x_m 75000000000000.0)
    (- (/ (fma y x_m x_m) z) x_m)
    (* (- (- y z) -1.0) (/ 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 (x_m <= 75000000000000.0) {
		tmp = (fma(y, x_m, x_m) / z) - x_m;
	} else {
		tmp = ((y - z) - -1.0) * (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 (x_m <= 75000000000000.0)
		tmp = Float64(Float64(fma(y, x_m, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(Float64(y - z) - -1.0) * Float64(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[LessEqual[x$95$m, 75000000000000.0], N[(N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e13

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if 7.5e13 < x

   1. Initial program 88.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. 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. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. mul-1-neg N/A

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

      14. associate-+r+ N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower--.f64 N/A

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

      25. lift--.f64 N/A

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

      26. lower-/.f64 89.0

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

   3. Applied rewrites 89.0%

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

Alternative 2: 95.8% accurate, 1.1× speedup

Math

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

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 (- (/ (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) {
	return x_s * ((fma(y, x_m, 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(fma(y, x_m, 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[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 95.8

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

4. Applied rewrites 95.8%

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

Alternative 3: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{y \cdot x\_m}{z} - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3300000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- (/ (* y x_m) z) x_m)))
   (*
    x_s
    (if (<= z -3300000000000.0)
      t_0
      (if (<= z 1.0) (/ (fma y x_m x_m) z) t_0)))))

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_0 = ((y * x_m) / z) - x_m;
	double tmp;
	if (z <= -3300000000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = fma(y, x_m, x_m) / z;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(Float64(y * x_m) / z) - x_m)
	tmp = 0.0
	if (z <= -3300000000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(fma(y, x_m, x_m) / z);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3300000000000.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e12 or 1 < z

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if -3.3e12 < z < 1

   1. Initial program 88.1%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.0

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

   4. Applied rewrites 61.0%

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

Alternative 4: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+23}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 -1.38e+23)
    (- x_m)
    (if (<= z 4e+49) (/ (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 <= -1.38e+23) {
		tmp = -x_m;
	} else if (z <= 4e+49) {
		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 <= -1.38e+23)
		tmp = Float64(-x_m);
	elseif (z <= 4e+49)
		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, -1.38e+23], (-x$95$m), If[LessEqual[z, 4e+49], N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], (-x$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.38e23 or 3.99999999999999979e49 < z

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.7

         \[\leadsto -x \]

   4. Applied rewrites 38.7%

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

   if -1.38e23 < z < 3.99999999999999979e49

   1. Initial program 88.1%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.0

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

   4. Applied rewrites 61.0%

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

Alternative 5: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{y \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* y x_m) z)))
   (* x_s (if (<= y -8e+87) t_0 (if (<= y 5.2e+93) (- (/ x_m z) x_m) t_0)))))

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_0 = (y * x_m) / z;
	double tmp;
	if (y <= -8e+87) {
		tmp = t_0;
	} else if (y <= 5.2e+93) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y * x_m) / z
    if (y <= (-8d+87)) then
        tmp = t_0
    else if (y <= 5.2d+93) then
        tmp = (x_m / z) - x_m
    else
        tmp = t_0
    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_0 = (y * x_m) / z;
	double tmp;
	if (y <= -8e+87) {
		tmp = t_0;
	} else if (y <= 5.2e+93) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = t_0;
	}
	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_0 = (y * x_m) / z
	tmp = 0
	if y <= -8e+87:
		tmp = t_0
	elif y <= 5.2e+93:
		tmp = (x_m / z) - x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(y * x_m) / z)
	tmp = 0.0
	if (y <= -8e+87)
		tmp = t_0;
	elseif (y <= 5.2e+93)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = t_0;
	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_0 = (y * x_m) / z;
	tmp = 0.0;
	if (y <= -8e+87)
		tmp = t_0;
	elseif (y <= 5.2e+93)
		tmp = (x_m / z) - x_m;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -8e+87], t$95$0, If[LessEqual[y, 5.2e+93], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999997e87 or 5.19999999999999999e93 < y

   1. Initial program 88.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.7

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

   4. Applied rewrites 38.7%

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

   if -7.9999999999999997e87 < y < 5.19999999999999999e93

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 65.1

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

   7. Applied rewrites 65.1%

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

Alternative 6: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (/ x_m z) y)))
   (* x_s (if (<= y -8e+87) t_0 (if (<= y 5.2e+93) (- (/ x_m z) x_m) t_0)))))

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_0 = (x_m / z) * y;
	double tmp;
	if (y <= -8e+87) {
		tmp = t_0;
	} else if (y <= 5.2e+93) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_m / z) * y
    if (y <= (-8d+87)) then
        tmp = t_0
    else if (y <= 5.2d+93) then
        tmp = (x_m / z) - x_m
    else
        tmp = t_0
    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_0 = (x_m / z) * y;
	double tmp;
	if (y <= -8e+87) {
		tmp = t_0;
	} else if (y <= 5.2e+93) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = t_0;
	}
	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_0 = (x_m / z) * y
	tmp = 0
	if y <= -8e+87:
		tmp = t_0
	elif y <= 5.2e+93:
		tmp = (x_m / z) - x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m / z) * y)
	tmp = 0.0
	if (y <= -8e+87)
		tmp = t_0;
	elseif (y <= 5.2e+93)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = t_0;
	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_0 = (x_m / z) * y;
	tmp = 0.0;
	if (y <= -8e+87)
		tmp = t_0;
	elseif (y <= 5.2e+93)
		tmp = (x_m / z) - x_m;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -8e+87], t$95$0, If[LessEqual[y, 5.2e+93], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999997e87 or 5.19999999999999999e93 < y

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   if -7.9999999999999997e87 < y < 5.19999999999999999e93

   1. Initial program 88.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 65.1

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

   7. Applied rewrites 65.1%

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

Alternative 7: 65.1% accurate, 1.8× 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 88.1%

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

2. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. *-rgt-identity N/A

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

   11. lower-fma.f64 95.8

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. lift-/.f64 65.1

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

7. Applied rewrites 65.1%

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

Alternative 8: 38.7% accurate, 6.4× 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 88.1%

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

2. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 38.7

      \[\leadsto -x \]

4. Applied rewrites 38.7%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64
  (/ (* x (+ (- y z) 1.0)) z))