Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \sin y + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}

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 * sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x * math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sin y + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}

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 * sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x * math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y, z, \sin y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (cos y) z (* (sin y) x)))

C

double code(double x, double y, double z) {
	return fma(cos(y), z, (sin(y) * x));
}

Julia

function code(x, y, z)
	return fma(cos(y), z, Float64(sin(y) * x))
end

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-167} \lor \neg \left(x \leq 320\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9e-167) (not (<= x 320.0)))
   (fma 1.0 z (* (sin y) x))
   (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e-167) || !(x <= 320.0)) {
		tmp = fma(1.0, z, (sin(y) * x));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9e-167) || !(x <= 320.0))
		tmp = fma(1.0, z, Float64(sin(y) * x));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9e-167], N[Not[LessEqual[x, 320.0]], $MachinePrecision]], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000002e-167 or 320 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.0%

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

   if -9.0000000000000002e-167 < x < 320

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 92.0

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\cos y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-167} \lor \neg \left(x \leq 320\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;y \leq -0.009:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= y -0.009)
     t_0
     (if (<= y 5.1e-9)
       (fma (fma (* z y) -0.5 x) y z)
       (if (<= y 7e+144) (* x (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (y <= -0.009) {
		tmp = t_0;
	} else if (y <= 5.1e-9) {
		tmp = fma(fma((z * y), -0.5, x), y, z);
	} else if (y <= 7e+144) {
		tmp = x * sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (y <= -0.009)
		tmp = t_0;
	elseif (y <= 5.1e-9)
		tmp = fma(fma(Float64(z * y), -0.5, x), y, z);
	elseif (y <= 7e+144)
		tmp = Float64(x * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -0.009], t$95$0, If[LessEqual[y, 5.1e-9], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + x), $MachinePrecision] * y + z), $MachinePrecision], If[LessEqual[y, 7e+144], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00899999999999999932 or 6.9999999999999996e144 < y

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 57.2

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -0.00899999999999999932 < y < 5.10000000000000017e-9

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + y \cdot \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{z - \left(\mathsf{neg}\left(y\right)\right) \cdot \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \]

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

         \[\leadsto \color{blue}{z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \cdot \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \cdot \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) + z} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)\right)} + z \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) + z \]

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

         \[\leadsto \color{blue}{\left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} + z \]

      7. remove-double-neg N/A

         \[\leadsto \left(x + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot \color{blue}{y} + z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x + \frac{-1}{2} \cdot \left(y \cdot z\right), y, z\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right) + x}, y, z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot \frac{-1}{2}} + x, y, z\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot z, \frac{-1}{2}, x\right)}, y, z\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{-1}{2}, x\right), y, z\right) \]

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot y}, -0.5, x\right), y, z\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, x\right), y, z\right)} \]

   if 5.10000000000000017e-9 < y < 6.9999999999999996e144

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto {\left(z \cdot \cos y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x \cdot \sin y \]

      5. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(z \cdot \cos y\right)}^{2}}} + x \cdot \sin y \]

      6. pow2 N/A

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

      7. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{z \cdot \cos y} \cdot \sqrt{z \cdot \cos y}} + x \cdot \sin y \]

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 49.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 49.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 49.7

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

   4. Applied rewrites 49.7%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-1 \cdot \frac{x \cdot \sin y}{z} + \cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(-1 \cdot \frac{x \cdot \sin y}{z} + \cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-1 \cdot z\right) \cdot \color{blue}{\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2} + -1 \cdot \frac{x \cdot \sin y}{z}\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \left(-1 \cdot z\right) \cdot \left(-1 \cdot \frac{x \cdot \sin y}{z}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot z\right) \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos y\right)} + \left(-1 \cdot z\right) \cdot \left(-1 \cdot \frac{x \cdot \sin y}{z}\right) \]

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-lft-out N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 91.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \sin y} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \sin y} \]

      2. lower-sin.f64 74.2

         \[\leadsto x \cdot \color{blue}{\sin y} \]

   10. Applied rewrites 74.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.009:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 12200000000000\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.011) (not (<= y 12200000000000.0)))
   (* (cos y) z)
   (fma (fma (fma -0.16666666666666666 (* y x) (* -0.5 z)) y x) y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.011) || !(y <= 12200000000000.0)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, (y * x), (-0.5 * z)), y, x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.011) || !(y <= 12200000000000.0))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(fma(fma(-0.16666666666666666, Float64(y * x), Float64(-0.5 * z)), y, x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.011], N[Not[LessEqual[y, 12200000000000.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(y * x), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 1.22e13 < y

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 52.6

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -0.010999999999999999 < y < 1.22e13

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right) + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right) \cdot y} + z \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right), y, z\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right) + x}, y, z\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right) \cdot y} + x, y, z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right), y, x\right)}, y, z\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot y\right) + \frac{-1}{2} \cdot z}, y, x\right), y, z\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot y, \frac{-1}{2} \cdot z\right)}, y, x\right), y, z\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot x}, \frac{-1}{2} \cdot z\right), y, x\right), y, z\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot x}, \frac{-1}{2} \cdot z\right), y, x\right), y, z\right) \]

      11. lower-*.f64 98.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, \color{blue}{-0.5 \cdot z}\right), y, x\right), y, z\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 12200000000000\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+156} \lor \neg \left(x \leq 1.35 \cdot 10^{+209}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.05e+156) (not (<= x 1.35e+209))) (* x y) (* 1.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+156) || !(x <= 1.35e+209)) {
		tmp = x * y;
	} else {
		tmp = 1.0 * z;
	}
	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) :: tmp
    if ((x <= (-1.05d+156)) .or. (.not. (x <= 1.35d+209))) then
        tmp = x * y
    else
        tmp = 1.0d0 * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+156) || !(x <= 1.35e+209)) {
		tmp = x * y;
	} else {
		tmp = 1.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.05e+156) or not (x <= 1.35e+209):
		tmp = x * y
	else:
		tmp = 1.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.05e+156) || !(x <= 1.35e+209))
		tmp = Float64(x * y);
	else
		tmp = Float64(1.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.05e+156) || ~((x <= 1.35e+209)))
		tmp = x * y;
	else
		tmp = 1.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.05e+156], N[Not[LessEqual[x, 1.35e+209]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(1.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999991e156 or 1.35e209 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.1%

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

   if -1.04999999999999991e156 < x < 1.35e209

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 72.0

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 45.4%

      \[\leadsto 1 \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+156} \lor \neg \left(x \leq 1.35 \cdot 10^{+209}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 30.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y x z))

C

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

Julia

function code(x, y, z)
	return fma(y, x, z)
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Final simplification 54.5%

   \[\leadsto \mathsf{fma}\left(y, x, z\right) \]
7. Add Preprocessing

Alternative 7: 16.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 19.4%

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

9. Final simplification 19.4%

   \[\leadsto x \cdot y \]
10. Add Preprocessing

Alternative 8: 4.7% accurate, 71.3× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   3. lower-cos.f64 60.2

      \[\leadsto \color{blue}{\cos y} \cdot z \]

5. Applied rewrites 60.2%

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

7. Applied rewrites 2.3%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 4.6%

    \[\leadsto -z \]

11. Final simplification 4.6%

    \[\leadsto -z \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))