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

Percentage Accurate: 88.2% → 99.8%

Time: 3.2s

Alternatives: 10

Speedup: 0.9×

• 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: 88.2% 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, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{1 - z}{z} + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e+39)
    (- (/ (fma y x_m x_m) z) x_m)
    (* x_m (+ (/ (- 1.0 z) z) (/ y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1e+39) {
		tmp = (fma(y, x_m, x_m) / z) - x_m;
	} else {
		tmp = x_m * (((1.0 - z) / z) + (y / z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1e+39)
		tmp = Float64(Float64(fma(y, x_m, x_m) / z) - x_m);
	else
		tmp = Float64(x_m * Float64(Float64(Float64(1.0 - z) / z) + 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+39], N[(N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m * N[(N[(N[(1.0 - z), $MachinePrecision] / z), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e38

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999994e38 < x

   1. Initial program 73.9%

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

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. associate-+r- N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. div-add-rev N/A

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

      12. associate-/l* N/A

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

      13. associate-/l* N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y}{z} - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (- (* x_m (/ y z)) x_m)))
   (* x_s (if (<= z -7.4) t_0 (if (<= z 0.85) (/ (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 = (x_m * (y / z)) - x_m;
	double tmp;
	if (z <= -7.4) {
		tmp = t_0;
	} else if (z <= 0.85) {
		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(x_m * Float64(y / z)) - x_m)
	tmp = 0.0
	if (z <= -7.4)
		tmp = t_0;
	elseif (z <= 0.85)
		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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -7.4], t$95$0, If[LessEqual[z, 0.85], 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 < -7.4000000000000004 or 0.849999999999999978 < z

   1. Initial program 76.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 99.0

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

   8. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 95.4

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

   10. Applied rewrites 95.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if -7.4000000000000004 < z < 0.849999999999999978

   1. Initial program 99.9%

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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 3: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+32}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= z -5.5e+32)
    (- x_m)
    (if (<= z 3.4e+40) (/ (fma y x_m x_m) z) (- (/ 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 <= -5.5e+32) {
		tmp = -x_m;
	} else if (z <= 3.4e+40) {
		tmp = fma(y, x_m, x_m) / z;
	} 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 (z <= -5.5e+32)
		tmp = Float64(-x_m);
	elseif (z <= 3.4e+40)
		tmp = Float64(fma(y, x_m, x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) - 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, -5.5e+32], (-x$95$m), If[LessEqual[z, 3.4e+40], N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.49999999999999984e32

   1. Initial program 73.4%

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

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

      2. lower-neg.f64 77.6

         \[\leadsto -x \]

   5. Applied rewrites 77.6%

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

   if -5.49999999999999984e32 < z < 3.39999999999999989e40

   1. Initial program 99.5%

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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 3.39999999999999989e40 < z

   1. Initial program 73.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.1%

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

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -1.85e+25) t_0 (if (<= y 9.6e+95) (- (/ 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 <= -1.85e+25) {
		tmp = t_0;
	} else if (y <= 9.6e+95) {
		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 <= (-1.85d+25)) then
        tmp = t_0
    else if (y <= 9.6d+95) 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 <= -1.85e+25) {
		tmp = t_0;
	} else if (y <= 9.6e+95) {
		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 <= -1.85e+25:
		tmp = t_0
	elif y <= 9.6e+95:
		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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -1.85e+25)
		tmp = t_0;
	elseif (y <= 9.6e+95)
		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 <= -1.85e+25)
		tmp = t_0;
	elseif (y <= 9.6e+95)
		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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.85e+25], t$95$0, If[LessEqual[y, 9.6e+95], 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 < -1.8499999999999999e25 or 9.6000000000000002e95 < y

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.8499999999999999e25 < y < 9.6000000000000002e95

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.0%

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

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) - -1}{z} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e+39)
    (- (/ (fma y x_m x_m) z) x_m)
    (* (/ (- (- y z) -1.0) 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 (x_m <= 1e+39) {
		tmp = (fma(y, x_m, x_m) / z) - x_m;
	} else {
		tmp = (((y - z) - -1.0) / 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 (x_m <= 1e+39)
		tmp = Float64(Float64(fma(y, x_m, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(Float64(Float64(y - z) - -1.0) / z) * 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, 1e+39], N[(N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(N[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e38

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999994e38 < x

   1. Initial program 73.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. 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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 6: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z\right) - -1\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 2.8e+39)
    (- (/ (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 <= 2.8e+39) {
		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 <= 2.8e+39)
		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, 2.8e+39], 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 < 2.80000000000000001e39

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2.80000000000000001e39 < x

   1. Initial program 73.9%

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 7: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= y 1.2e+16) (- (/ (fma y x_m x_m) z) x_m) (- (* 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 <= 1.2e+16) {
		tmp = (fma(y, x_m, x_m) / z) - x_m;
	} else {
		tmp = (y * (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 <= 1.2e+16)
		tmp = Float64(Float64(fma(y, x_m, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(y * Float64(x_m / z)) - 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[y, 1.2e+16], N[(N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e16

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 1.2e16 < y

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
   4. 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-rgt-in N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 92.5

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

   8. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 95.5

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

   10. Applied rewrites 95.5%

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

Alternative 8: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.285:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 -0.285) (- x_m) (if (<= z 2.95e-5) (/ 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 <= -0.285) {
		tmp = -x_m;
	} else if (z <= 2.95e-5) {
		tmp = x_m / z;
	} else {
		tmp = -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 (z <= (-0.285d0)) then
        tmp = -x_m
    else if (z <= 2.95d-5) then
        tmp = x_m / z
    else
        tmp = -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 (z <= -0.285) {
		tmp = -x_m;
	} else if (z <= 2.95e-5) {
		tmp = x_m / z;
	} else {
		tmp = -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 z <= -0.285:
		tmp = -x_m
	elif z <= 2.95e-5:
		tmp = 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 <= -0.285)
		tmp = Float64(-x_m);
	elseif (z <= 2.95e-5)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(-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 (z <= -0.285)
		tmp = -x_m;
	elseif (z <= 2.95e-5)
		tmp = x_m / z;
	else
		tmp = -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[LessEqual[z, -0.285], (-x$95$m), If[LessEqual[z, 2.95e-5], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -0.284999999999999976 or 2.9499999999999999e-5 < z

   1. Initial program 76.7%

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

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

      2. lower-neg.f64 73.3

         \[\leadsto -x \]

   5. Applied rewrites 73.3%

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

   if -0.284999999999999976 < z < 2.9499999999999999e-5

   1. Initial program 99.9%

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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.6%

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

Alternative 9: 65.4% 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 88.2%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-1 \cdot x + \frac{x \cdot \left(1 + y\right)}{z}} \]
4. 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-rgt-in N/A

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

   9. *-lft-identity N/A

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

   10. lower-fma.f64 95.6

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 65.4%

   \[\leadsto \frac{x}{z} - x \]
9. Add Preprocessing

Alternative 10: 38.3% 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 88.2%

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

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

   2. lower-neg.f64 38.3

      \[\leadsto -x \]

5. Applied rewrites 38.3%

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

Developer Target 1: 99.4% 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 2025091 
(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))