Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Percentage Accurate: 96.4% → 99.8%

Time: 2.8s

Alternatives: 9

Speedup: 0.8×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * 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 * (1.0d0 - ((1.0d0 - y) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot x\_m, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot z, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 5e-56)
    (fma (* (- y 1.0) x_m) z x_m)
    (fma (* (- y 1.0) z) x_m 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 <= 5e-56) {
		tmp = fma(((y - 1.0) * x_m), z, x_m);
	} else {
		tmp = fma(((y - 1.0) * z), x_m, 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 <= 5e-56)
		tmp = fma(Float64(Float64(y - 1.0) * x_m), z, x_m);
	else
		tmp = fma(Float64(Float64(y - 1.0) * z), x_m, 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, 5e-56], N[(N[(N[(y - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999997e-56

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 95.9

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

   3. Applied rewrites 95.9%

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

   if 4.99999999999999997e-56 < x

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 96.4

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

   3. Applied rewrites 96.4%

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

Alternative 2: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot y, z, x\_m\right)\\ t_1 := 1 - \left(1 - y\right) \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5000000:\\ \;\;\;\;x\_m \cdot \left(z \cdot y - z\right)\\ \mathbf{elif}\;t\_1 \leq 400:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(y - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* x_m y) z x_m)) (t_1 (- 1.0 (* (- 1.0 y) z))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      t_0
      (if (<= t_1 -5000000.0)
        (* x_m (- (* z y) z))
        (if (<= t_1 400.0)
          (* x_m (- 1.0 z))
          (if (<= t_1 2e+294) (* x_m (* z (- y 1.0))) 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 = fma((x_m * y), z, x_m);
	double t_1 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -5000000.0) {
		tmp = x_m * ((z * y) - z);
	} else if (t_1 <= 400.0) {
		tmp = x_m * (1.0 - z);
	} else if (t_1 <= 2e+294) {
		tmp = x_m * (z * (y - 1.0));
	} 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 = fma(Float64(x_m * y), z, x_m)
	t_1 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -5000000.0)
		tmp = Float64(x_m * Float64(Float64(z * y) - z));
	elseif (t_1 <= 400.0)
		tmp = Float64(x_m * Float64(1.0 - z));
	elseif (t_1 <= 2e+294)
		tmp = Float64(x_m * Float64(z * Float64(y - 1.0)));
	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 * y), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -5000000.0], N[(x$95$m * N[(N[(z * y), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 400.0], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+294], N[(x$95$m * N[(z * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -inf.0 or 2.00000000000000013e294 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -5e6

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.1

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

   4. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. add-flip N/A

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

      7. distribute-lft-neg-out N/A

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

      8. remove-double-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - 1 \cdot z\right) \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \left(y \cdot z - z\right) \]

      10. lower--.f64 59.1

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

      11. lift-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot z - z\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \left(z \cdot y - z\right) \]

      13. lower-*.f64 59.1

          \[\leadsto x \cdot \left(z \cdot y - z\right) \]

   6. Applied rewrites 59.1%

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

   if -5e6 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 400

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 66.6%

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

   if 400 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 2.00000000000000013e294

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.1

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

   4. Applied rewrites 59.1%

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

Alternative 3: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot y, z, x\_m\right)\\ t_1 := x\_m \cdot \left(z \cdot \left(y - 1\right)\right)\\ t_2 := 1 - \left(1 - y\right) \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 400:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+294}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* x_m y) z x_m))
        (t_1 (* x_m (* z (- y 1.0))))
        (t_2 (- 1.0 (* (- 1.0 y) z))))
   (*
    x_s
    (if (<= t_2 (- INFINITY))
      t_0
      (if (<= t_2 -5000000.0)
        t_1
        (if (<= t_2 400.0) (* x_m (- 1.0 z)) (if (<= t_2 2e+294) t_1 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 = fma((x_m * y), z, x_m);
	double t_1 = x_m * (z * (y - 1.0));
	double t_2 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= -5000000.0) {
		tmp = t_1;
	} else if (t_2 <= 400.0) {
		tmp = x_m * (1.0 - z);
	} else if (t_2 <= 2e+294) {
		tmp = t_1;
	} 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 = fma(Float64(x_m * y), z, x_m)
	t_1 = Float64(x_m * Float64(z * Float64(y - 1.0)))
	t_2 = Float64(1.0 - Float64(Float64(1.0 - y) * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= -5000000.0)
		tmp = t_1;
	elseif (t_2 <= 400.0)
		tmp = Float64(x_m * Float64(1.0 - z));
	elseif (t_2 <= 2e+294)
		tmp = t_1;
	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 * y), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(z * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -5000000.0], t$95$1, If[LessEqual[t$95$2, 400.0], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+294], t$95$1, t$95$0]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -inf.0 or 2.00000000000000013e294 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z))

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if -inf.0 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < -5e6 or 400 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 2.00000000000000013e294

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.1

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

   4. Applied rewrites 59.1%

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

   if -5e6 < (-.f64 #s(literal 1 binary64) (*.f64 (-.f64 #s(literal 1 binary64) y) z)) < 400

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 66.6%

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

Alternative 4: 95.9% accurate, 1.1× speedup

Math

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

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 (fma (* (- y 1.0) 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 - 1.0) * 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 * fma(Float64(Float64(y - 1.0) * 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 - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * z + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower--.f64 95.9

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

3. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - 1\right) \cdot x, z, x\right)} \]
4. Add Preprocessing

Alternative 5: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot y, z, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-z, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (* x_m y) z x_m)))
   (* x_s (if (<= y -600.0) t_0 (if (<= y 1.0) (fma (- z) x_m 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 = fma((x_m * y), z, x_m);
	double tmp;
	if (y <= -600.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(-z, x_m, 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 = fma(Float64(x_m * y), z, x_m)
	tmp = 0.0
	if (y <= -600.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(-z), x_m, x_m);
	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 * y), $MachinePrecision] * z + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -600.0], t$95$0, If[LessEqual[y, 1.0], N[((-z) * x$95$m + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -600 or 1 < y

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if -600 < y < 1

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 66.6

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

   6. Applied rewrites 66.6%

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

Alternative 6: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(-z, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= y -1.2e+81) t_0 (if (<= y 8.2e+68) (fma (- z) x_m 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 * (y * z);
	double tmp;
	if (y <= -1.2e+81) {
		tmp = t_0;
	} else if (y <= 8.2e+68) {
		tmp = fma(-z, x_m, 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(x_m * Float64(y * z))
	tmp = 0.0
	if (y <= -1.2e+81)
		tmp = t_0;
	elseif (y <= 8.2e+68)
		tmp = fma(Float64(-z), x_m, x_m);
	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[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.2e+81], t$95$0, If[LessEqual[y, 8.2e+68], N[((-z) * x$95$m + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999995e81 or 8.1999999999999998e68 < y

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

   if -1.19999999999999995e81 < y < 8.1999999999999998e68

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 66.6

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

   6. Applied rewrites 66.6%

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

Alternative 7: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(y \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;x\_m \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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_s (if (<= y -1.2e+81) t_0 (if (<= y 8.2e+68) (* x_m (- 1.0 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);
	double tmp;
	if (y <= -1.2e+81) {
		tmp = t_0;
	} else if (y <= 8.2e+68) {
		tmp = x_m * (1.0 - z);
	} 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 * (y * z)
    if (y <= (-1.2d+81)) then
        tmp = t_0
    else if (y <= 8.2d+68) then
        tmp = x_m * (1.0d0 - z)
    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 * (y * z);
	double tmp;
	if (y <= -1.2e+81) {
		tmp = t_0;
	} else if (y <= 8.2e+68) {
		tmp = x_m * (1.0 - z);
	} 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 * (y * z)
	tmp = 0
	if y <= -1.2e+81:
		tmp = t_0
	elif y <= 8.2e+68:
		tmp = x_m * (1.0 - 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(x_m * Float64(y * z))
	tmp = 0.0
	if (y <= -1.2e+81)
		tmp = t_0;
	elseif (y <= 8.2e+68)
		tmp = Float64(x_m * Float64(1.0 - z));
	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 * (y * z);
	tmp = 0.0;
	if (y <= -1.2e+81)
		tmp = t_0;
	elseif (y <= 8.2e+68)
		tmp = x_m * (1.0 - z);
	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[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.2e+81], t$95$0, If[LessEqual[y, 8.2e+68], N[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999995e81 or 8.1999999999999998e68 < y

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

   if -1.19999999999999995e81 < y < 8.1999999999999998e68

   1. Initial program 96.4%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 66.6%

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

Alternative 8: 66.6% accurate, 1.8× speedup

Math

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

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 (- 1.0 z))))

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 * (1.0 - z));
}

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 * (1.0d0 - z))
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 * (1.0 - z));
}

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 * (1.0 - z))

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 * Float64(1.0 - z)))
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 * (1.0 - z));
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[(x$95$m * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

2. Taylor expanded in y around 0

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

4. Applied rewrites 66.6%

   \[\leadsto x \cdot \left(1 - \color{blue}{z}\right) \]
5. Add Preprocessing

Alternative 9: 39.3% accurate, 3.1× speedup

Math

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

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 1.0)))

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 * 1.0);
}

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 * 1.0d0)
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 * 1.0);
}

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 * 1.0)

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 * 1.0))
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 * 1.0);
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[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

2. Taylor expanded in y around 0

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

4. Applied rewrites 66.6%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 39.3%

   \[\leadsto x \cdot \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64
  (* x (- 1.0 (* (- 1.0 y) z))))