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

Percentage Accurate: 95.8% → 100.0%

Time: 37.3s

Alternatives: 11

Speedup: 1.0×

• 78.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (* x (- 1 (* (- 1 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 - N[(N[(1 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (* x (- 1 (* (- 1 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 - N[(N[(1 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\mathsf{134\_z0z1z2z3z4}\left(x, z, y, \left(z - 1\right), 1\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (134-z0z1z2z3z4 x z y (- z 1) 1))

Derivation

1. Initial program 95.8%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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. add-flip N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt1-in N/A

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

   11. *-commutative N/A

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

   12. associate--r+ N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-rgt-out-- N/A

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

   17. *-commutative N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. distribute-lft-neg-in N/A

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

   20. lower-134-z0z1z2z3z4 N/A

       \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(\mathsf{neg}\left(z\right)\right), y\right)} \]

   21. lower--.f64 N/A

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \color{blue}{\left(1 - z\right)}, \left(\mathsf{neg}\left(z\right)\right), y\right) \]

   22. lower-neg.f64 100.0%

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \color{blue}{\left(-z\right)}, y\right) \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(-z\right), y\right)} \]
4. Step-by-step derivation

   1. lift-134-z0z1z2z3z4 N/A

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

   2. lift-neg.f64 N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-rgt-identity N/A

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

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

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

   8. *-lft-identity N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower-134-z0z1z2z3z4 N/A

       \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, z, y, \left(z - 1\right), 1\right)} \]

   13. lower--.f64 100.0%

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, z, y, \color{blue}{\left(z - 1\right)}, 1\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, z, y, \left(z - 1\right), 1\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq \frac{2993155353253689}{2993155353253689176481146537402947624255349848014848}:\\ \;\;\;\;\left|x\right| - \left(\left(1 - y\right) \cdot \left|x\right|\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| - \left(z \cdot \left(1 - y\right)\right) \cdot \left|x\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1 x)
 (if (<=
      (fabs x)
      2993155353253689/2993155353253689176481146537402947624255349848014848)
   (- (fabs x) (* (* (- 1 y) (fabs x)) z))
   (- (fabs x) (* (* z (- 1 y)) (fabs x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(x) <= 1e-36) {
		tmp = fabs(x) - (((1.0 - y) * fabs(x)) * z);
	} else {
		tmp = fabs(x) - ((z * (1.0 - y)) * fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs(x) <= 1e-36) {
		tmp = Math.abs(x) - (((1.0 - y) * Math.abs(x)) * z);
	} else {
		tmp = Math.abs(x) - ((z * (1.0 - y)) * Math.abs(x));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs(x) <= 1e-36:
		tmp = math.fabs(x) - (((1.0 - y) * math.fabs(x)) * z)
	else:
		tmp = math.fabs(x) - ((z * (1.0 - y)) * math.fabs(x))
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(x) <= 1e-36)
		tmp = Float64(abs(x) - Float64(Float64(Float64(1.0 - y) * abs(x)) * z));
	else
		tmp = Float64(abs(x) - Float64(Float64(z * Float64(1.0 - y)) * abs(x)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs(x) <= 1e-36)
		tmp = abs(x) - (((1.0 - y) * abs(x)) * z);
	else
		tmp = abs(x) - ((z * (1.0 - y)) * abs(x));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 2993155353253689/2993155353253689176481146537402947624255349848014848], N[(N[Abs[x], $MachinePrecision] - N[(N[(N[(1 - y), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] - N[(N[(z * N[(1 - y), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e-37

   1. Initial program 95.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. add-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-commutative N/A

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

      12. associate--r+ N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-out-- N/A

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

      17. *-commutative N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-134-z0z1z2z3z4 N/A

          \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(\mathsf{neg}\left(z\right)\right), y\right)} \]

      21. lower--.f64 N/A

          \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \color{blue}{\left(1 - z\right)}, \left(\mathsf{neg}\left(z\right)\right), y\right) \]

      22. lower-neg.f64 100.0%

          \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \color{blue}{\left(-z\right)}, y\right) \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(-z\right), y\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. lift-neg.f64 N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--r- N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. distribute-rgt-in N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 96.2%

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

   7. Applied rewrites 96.2%

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

   if 9.9999999999999994e-37 < x

   1. Initial program 95.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. sub-flip-reverse N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.8%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.8%

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

   3. Applied rewrites 95.8%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (- x (* (- 1 y) (* z x))))

C

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

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 - y) * (z * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(1 - y), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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. add-flip N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt1-in N/A

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

   11. *-commutative N/A

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

   12. associate--r+ N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-rgt-out-- N/A

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

   17. *-commutative N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. distribute-lft-neg-in N/A

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

   20. lower-134-z0z1z2z3z4 N/A

       \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(\mathsf{neg}\left(z\right)\right), y\right)} \]

   21. lower--.f64 N/A

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \color{blue}{\left(1 - z\right)}, \left(\mathsf{neg}\left(z\right)\right), y\right) \]

   22. lower-neg.f64 100.0%

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \color{blue}{\left(-z\right)}, y\right) \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(-z\right), y\right)} \]
4. Step-by-step derivation

   1. lift-134-z0z1z2z3z4 N/A

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

   2. lift-neg.f64 N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-lft-identity N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--r- N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. distribute-rgt-in N/A

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

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

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

   12. *-lft-identity N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 95.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 98.0%

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

7. Applied rewrites 98.0%

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

Alternative 4: 95.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (- x (* (* z (- 1 y)) x)))

C

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

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 - ((z * (1.0d0 - y)) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(z * N[(1 - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

   \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
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. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. sub-flip-reverse N/A

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

   8. lower--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 95.8%

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 95.8%

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

3. Applied rewrites 95.8%

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

Alternative 5: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(z \cdot y - -1\right) \cdot x\\ \mathbf{if}\;1 - y \leq -500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - y \leq 2:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* (- (* z y) -1) x)))
  (if (<= (- 1 y) -500000) t_0 (if (<= (- 1 y) 2) (- x (* z x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z * y) - -1.0) * x;
	double tmp;
	if ((1.0 - y) <= -500000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2.0) {
		tmp = x - (z * x);
	} 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 = ((z * y) - (-1.0d0)) * x
    if ((1.0d0 - y) <= (-500000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - y) <= 2.0d0) then
        tmp = x - (z * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z * y) - -1.0) * x;
	double tmp;
	if ((1.0 - y) <= -500000.0) {
		tmp = t_0;
	} else if ((1.0 - y) <= 2.0) {
		tmp = x - (z * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z * y) - -1.0) * x
	tmp = 0
	if (1.0 - y) <= -500000.0:
		tmp = t_0
	elif (1.0 - y) <= 2.0:
		tmp = x - (z * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z * y) - -1.0) * x)
	tmp = 0.0
	if (Float64(1.0 - y) <= -500000.0)
		tmp = t_0;
	elseif (Float64(1.0 - y) <= 2.0)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z * y) - -1.0) * x;
	tmp = 0.0;
	if ((1.0 - y) <= -500000.0)
		tmp = t_0;
	elseif ((1.0 - y) <= 2.0)
		tmp = x - (z * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z * y), $MachinePrecision] - -1), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(1 - y), $MachinePrecision], -500000], t$95$0, If[LessEqual[N[(1 - y), $MachinePrecision], 2], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -5e5 or 2 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 95.8%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
   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. add-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-commutative N/A

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

      12. associate--r+ N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-out-- N/A

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

      17. *-commutative N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lower-134-z0z1z2z3z4 N/A

          \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(\mathsf{neg}\left(z\right)\right), y\right)} \]

      21. lower--.f64 N/A

          \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \color{blue}{\left(1 - z\right)}, \left(\mathsf{neg}\left(z\right)\right), y\right) \]

      22. lower-neg.f64 100.0%

          \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \color{blue}{\left(-z\right)}, y\right) \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, 1, \left(1 - z\right), \left(-z\right), y\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. lift-neg.f64 N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. *-lft-identity N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-134-z0z1z2z3z4 N/A

          \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, z, y, \left(z - 1\right), 1\right)} \]

      13. lower--.f64 100.0%

          \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, z, y, \color{blue}{\left(z - 1\right)}, 1\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x, z, y, \left(z - 1\right), 1\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, z, y, \color{blue}{-1}, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.1%

      \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x, z, y, \color{blue}{-1}, 1\right) \]
   9. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-rgt-identity 71.9%

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

   10. Applied rewrites 71.9%

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

   if -5e5 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 65.6%

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

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

      5. fp-cancel-sub-sign N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6%

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

   6. Applied rewrites 65.6%

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

Alternative 6: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;x \cdot \left(y \cdot z - z\right)\\ \mathbf{elif}\;t\_0 \leq 10000000:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(y - 1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* (- 1 y) z)))
  (if (<= t_0 -10000000000000)
    (* x (- (* y z) z))
    (if (<= t_0 10000000) (- x (* z x)) (* x (* z (- y 1)))))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = x * ((y * z) - z);
	} else if (t_0 <= 10000000.0) {
		tmp = x - (z * x);
	} else {
		tmp = x * (z * (y - 1.0));
	}
	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) * z
    if (t_0 <= (-10000000000000.0d0)) then
        tmp = x * ((y * z) - z)
    else if (t_0 <= 10000000.0d0) then
        tmp = x - (z * x)
    else
        tmp = x * (z * (y - 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = x * ((y * z) - z);
	} else if (t_0 <= 10000000.0) {
		tmp = x - (z * x);
	} else {
		tmp = x * (z * (y - 1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - y) * z
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = x * ((y * z) - z)
	elif t_0 <= 10000000.0:
		tmp = x - (z * x)
	else:
		tmp = x * (z * (y - 1.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= -10000000000000.0)
		tmp = Float64(x * Float64(Float64(y * z) - z));
	elseif (t_0 <= 10000000.0)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = Float64(x * Float64(z * Float64(y - 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	tmp = 0.0;
	if (t_0 <= -10000000000000.0)
		tmp = x * ((y * z) - z);
	elseif (t_0 <= 10000000.0)
		tmp = x - (z * x);
	else
		tmp = x * (z * (y - 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000000000], N[(x * N[(N[(y * z), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 10000000], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(z * N[(y - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -1e13

   1. Initial program 95.8%

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

   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 58.8%

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

   4. Applied rewrites 58.8%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot \left(y - 1\right)\right)} \]
   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-negate-rev N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. lift-neg.f64 N/A

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

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

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lift-neg.f64 N/A

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

      18. sub-flip N/A

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

      19. lower--.f64 58.8%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 58.8%

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

   6. Applied rewrites 58.8%

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

   if -1e13 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e7

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 65.6%

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

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

      5. fp-cancel-sub-sign N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6%

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

   6. Applied rewrites 65.6%

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

   if 1e7 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 95.8%

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

   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 58.8%

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

   4. Applied rewrites 58.8%

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

Alternative 7: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ t_1 := x \cdot \left(z \cdot \left(y - 1\right)\right)\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10000000:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* (- 1 y) z)) (t_1 (* x (* z (- y 1)))))
  (if (<= t_0 -10000000000000)
    t_1
    (if (<= t_0 10000000) (- x (* z x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double t_1 = x * (z * (y - 1.0));
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 10000000.0) {
		tmp = x - (z * x);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - y) * z
    t_1 = x * (z * (y - 1.0d0))
    if (t_0 <= (-10000000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 10000000.0d0) then
        tmp = x - (z * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double t_1 = x * (z * (y - 1.0));
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 10000000.0) {
		tmp = x - (z * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - y) * z
	t_1 = x * (z * (y - 1.0))
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = t_1
	elif t_0 <= 10000000.0:
		tmp = x - (z * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - y) * z)
	t_1 = Float64(x * Float64(z * Float64(y - 1.0)))
	tmp = 0.0
	if (t_0 <= -10000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 10000000.0)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	t_1 = x * (z * (y - 1.0));
	tmp = 0.0;
	if (t_0 <= -10000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 10000000.0)
		tmp = x - (z * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1 - y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(z * N[(y - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000000000], t$95$1, If[LessEqual[t$95$0, 10000000], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) y) z) < -1e13 or 1e7 < (*.f64 (-.f64 #s(literal 1 binary64) y) z)

   1. Initial program 95.8%

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

   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 58.8%

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

   4. Applied rewrites 58.8%

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

   if -1e13 < (*.f64 (-.f64 #s(literal 1 binary64) y) z) < 1e7

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 65.6%

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

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

      5. fp-cancel-sub-sign N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6%

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

   6. Applied rewrites 65.6%

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

Alternative 8: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -120:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 390000000000000010675228509652406409031340986106673798883456666958898901774965245340549120:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* x (* y z))))
  (if (<= y -120)
    t_0
    (if (<=
         y
         390000000000000010675228509652406409031340986106673798883456666958898901774965245340549120)
      (- x (* z x))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y * z);
	double tmp;
	if (y <= -120.0) {
		tmp = t_0;
	} else if (y <= 3.9e+89) {
		tmp = x - (z * x);
	} 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 = x * (y * z)
    if (y <= (-120.0d0)) then
        tmp = t_0
    else if (y <= 3.9d+89) then
        tmp = x - (z * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y * z);
	double tmp;
	if (y <= -120.0) {
		tmp = t_0;
	} else if (y <= 3.9e+89) {
		tmp = x - (z * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y * z)
	tmp = 0
	if y <= -120.0:
		tmp = t_0
	elif y <= 3.9e+89:
		tmp = x - (z * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -120.0)
		tmp = t_0;
	elseif (y <= 3.9e+89)
		tmp = Float64(x - Float64(z * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y * z);
	tmp = 0.0;
	if (y <= -120.0)
		tmp = t_0;
	elseif (y <= 3.9e+89)
		tmp = x - (z * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -120], t$95$0, If[LessEqual[y, 390000000000000010675228509652406409031340986106673798883456666958898901774965245340549120], N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -120 or 3.9000000000000001e89 < y

   1. Initial program 95.8%

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

   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 -120 < y < 3.9000000000000001e89

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 65.6%

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

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

      5. fp-cancel-sub-sign N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6%

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

   6. Applied rewrites 65.6%

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

Alternative 9: 65.6% accurate, 1.9× speedup

Math

\[x - z \cdot x \]

FPCore

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

C

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

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 - (z * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 65.6%

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

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

   5. fp-cancel-sub-sign N/A

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

   6. *-lft-identity N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 65.6%

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

6. Applied rewrites 65.6%

   \[\leadsto \color{blue}{x - z \cdot x} \]
7. Add Preprocessing

Alternative 10: 65.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (* x (- 1 z)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * N[(1 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 65.6%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64
  (* x (- 1 (* (- 1 y) z))))