Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, \frac{z}{x}, \cos y\right) \cdot x\\ \mathbf{elif}\;x \leq 8000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos y \cdot x}{z}, 1, \sin y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\sin y}{x}, \cos y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+34)
   (* (fma (sin y) (/ z x) (cos y)) x)
   (if (<= x 8000000.0)
     (* (fma (/ (* (cos y) x) z) 1.0 (sin y)) z)
     (* (fma z (/ (sin y) x) (cos y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+34) {
		tmp = fma(sin(y), (z / x), cos(y)) * x;
	} else if (x <= 8000000.0) {
		tmp = fma(((cos(y) * x) / z), 1.0, sin(y)) * z;
	} else {
		tmp = fma(z, (sin(y) / x), cos(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e+34)
		tmp = Float64(fma(sin(y), Float64(z / x), cos(y)) * x);
	elseif (x <= 8000000.0)
		tmp = Float64(fma(Float64(Float64(cos(y) * x) / z), 1.0, sin(y)) * z);
	else
		tmp = Float64(fma(z, Float64(sin(y) / x), cos(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+34], N[(N[(N[Sin[y], $MachinePrecision] * N[(z / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 8000000.0], N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] * 1.0 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999974e34

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -4.79999999999999974e34 < x < 8e6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 8e6 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

Alternative 3: 99.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sin.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, \frac{z}{x}, \cos y\right) \cdot x\\ \mathbf{elif}\;x \leq 15000000:\\ \;\;\;\;\mathsf{fma}\left(\cos y \cdot \frac{x}{z}, 1, \sin y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\sin y}{x}, \cos y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+34)
   (* (fma (sin y) (/ z x) (cos y)) x)
   (if (<= x 15000000.0)
     (* (fma (* (cos y) (/ x z)) 1.0 (sin y)) z)
     (* (fma z (/ (sin y) x) (cos y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+34) {
		tmp = fma(sin(y), (z / x), cos(y)) * x;
	} else if (x <= 15000000.0) {
		tmp = fma((cos(y) * (x / z)), 1.0, sin(y)) * z;
	} else {
		tmp = fma(z, (sin(y) / x), cos(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e+34)
		tmp = Float64(fma(sin(y), Float64(z / x), cos(y)) * x);
	elseif (x <= 15000000.0)
		tmp = Float64(fma(Float64(cos(y) * Float64(x / z)), 1.0, sin(y)) * z);
	else
		tmp = Float64(fma(z, Float64(sin(y) / x), cos(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+34], N[(N[(N[Sin[y], $MachinePrecision] * N[(z / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 15000000.0], N[(N[(N[(N[Cos[y], $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] * 1.0 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999974e34

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -4.79999999999999974e34 < x < 1.5e7

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.5e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 6: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{x}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, \frac{z}{x}, \cos y\right) \cdot x\\ \mathbf{elif}\;x \leq 11000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot t\_0\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_0, \cos y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) x)))
   (if (<= x -4.8e+34)
     (* (fma (sin y) (/ z x) (cos y)) x)
     (if (<= x 11000000.0)
       (* (fma (/ x z) (cos y) (* x t_0)) z)
       (* (fma z t_0 (cos y)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / x;
	double tmp;
	if (x <= -4.8e+34) {
		tmp = fma(sin(y), (z / x), cos(y)) * x;
	} else if (x <= 11000000.0) {
		tmp = fma((x / z), cos(y), (x * t_0)) * z;
	} else {
		tmp = fma(z, t_0, cos(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / x)
	tmp = 0.0
	if (x <= -4.8e+34)
		tmp = Float64(fma(sin(y), Float64(z / x), cos(y)) * x);
	elseif (x <= 11000000.0)
		tmp = Float64(fma(Float64(x / z), cos(y), Float64(x * t_0)) * z);
	else
		tmp = Float64(fma(z, t_0, cos(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -4.8e+34], N[(N[(N[Sin[y], $MachinePrecision] * N[(z / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 11000000.0], N[(N[(N[(x / z), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * t$95$0 + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999974e34

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -4.79999999999999974e34 < x < 1.1e7

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z}\right)}^{3} + {\sin y}^{3}}{\mathsf{fma}\left(\frac{\cos y \cdot x}{z}, \frac{\cos y \cdot x}{z}, {\sin y}^{2} - \frac{\cos y \cdot x}{z} \cdot \sin y\right)} \cdot z \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

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

   if 1.1e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+34} \lor \neg \left(x \leq 11000000\right):\\ \;\;\;\;\mathsf{fma}\left(z, t\_0, \cos y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot t\_0\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) x)))
   (if (or (<= x -5e+34) (not (<= x 11000000.0)))
     (* (fma z t_0 (cos y)) x)
     (* (fma (/ x z) (cos y) (* x t_0)) z))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / x;
	double tmp;
	if ((x <= -5e+34) || !(x <= 11000000.0)) {
		tmp = fma(z, t_0, cos(y)) * x;
	} else {
		tmp = fma((x / z), cos(y), (x * t_0)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / x)
	tmp = 0.0
	if ((x <= -5e+34) || !(x <= 11000000.0))
		tmp = Float64(fma(z, t_0, cos(y)) * x);
	else
		tmp = Float64(fma(Float64(x / z), cos(y), Float64(x * t_0)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]}, If[Or[LessEqual[x, -5e+34], N[Not[LessEqual[x, 11000000.0]], $MachinePrecision]], N[(N[(z * t$95$0 + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e34 or 1.1e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -4.9999999999999998e34 < x < 1.1e7

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z}\right)}^{3} + {\sin y}^{3}}{\mathsf{fma}\left(\frac{\cos y \cdot x}{z}, \frac{\cos y \cdot x}{z}, {\sin y}^{2} - \frac{\cos y \cdot x}{z} \cdot \sin y\right)} \cdot z \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+34} \lor \neg \left(x \leq 11000000\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\sin y}{x}, \cos y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-123} \lor \neg \left(x \leq 10000000\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\sin y \cdot z}{x}, \cos y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e-123) (not (<= x 10000000.0)))
   (* (fma 1.0 (/ (* (sin y) z) x) (cos y)) x)
   (* (fma (/ x z) (cos y) (* x (/ (sin y) x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-123) || !(x <= 10000000.0)) {
		tmp = fma(1.0, ((sin(y) * z) / x), cos(y)) * x;
	} else {
		tmp = fma((x / z), cos(y), (x * (sin(y) / x))) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e-123) || !(x <= 10000000.0))
		tmp = Float64(fma(1.0, Float64(Float64(sin(y) * z) / x), cos(y)) * x);
	else
		tmp = Float64(fma(Float64(x / z), cos(y), Float64(x * Float64(sin(y) / x))) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1e-123], N[Not[LessEqual[x, 10000000.0]], $MachinePrecision]], N[(N[(1.0 * N[(N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-123 or 1e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.0000000000000001e-123 < x < 1e7

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

   7. Applied rewrites 66.1%

      \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z}\right)}^{3} + {\sin y}^{3}}{\mathsf{fma}\left(\frac{\cos y \cdot x}{z}, \frac{\cos y \cdot x}{z}, {\sin y}^{2} - \frac{\cos y \cdot x}{z} \cdot \sin y\right)} \cdot z \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-123} \lor \neg \left(x \leq 10000000\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\sin y \cdot z}{x}, \cos y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+78} \lor \neg \left(z \leq 1.65 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\cos y}{z}, \frac{\sin y \cdot z}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e+78) (not (<= z 1.65e-87)))
   (* (fma (/ x z) (cos y) (* x (/ (sin y) x))) z)
   (* (fma z (/ (cos y) z) (/ (* (sin y) z) x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+78) || !(z <= 1.65e-87)) {
		tmp = fma((x / z), cos(y), (x * (sin(y) / x))) * z;
	} else {
		tmp = fma(z, (cos(y) / z), ((sin(y) * z) / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e+78) || !(z <= 1.65e-87))
		tmp = Float64(fma(Float64(x / z), cos(y), Float64(x * Float64(sin(y) / x))) * z);
	else
		tmp = Float64(fma(z, Float64(cos(y) / z), Float64(Float64(sin(y) * z) / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e+78], N[Not[LessEqual[z, 1.65e-87]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * N[(N[Cos[y], $MachinePrecision] / z), $MachinePrecision] + N[(N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000002e78 or 1.65e-87 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z} \cdot 1\right)}^{3} + {\sin y}^{3}}{\left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \left(\frac{\cos y \cdot x}{z} \cdot 1\right) + \left(\sin y \cdot \sin y - \left(\frac{\cos y \cdot x}{z} \cdot 1\right) \cdot \sin y\right)} \cdot z \]

   7. Applied rewrites 63.9%

      \[\leadsto \frac{{\left(\frac{\cos y \cdot x}{z}\right)}^{3} + {\sin y}^{3}}{\mathsf{fma}\left(\frac{\cos y \cdot x}{z}, \frac{\cos y \cdot x}{z}, {\sin y}^{2} - \frac{\cos y \cdot x}{z} \cdot \sin y\right)} \cdot z \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

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

   if -2.00000000000000002e78 < z < 1.65e-87

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(z \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot x \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 99.6

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

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(z, \frac{\cos y}{z}, \frac{\sin y \cdot z}{x}\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+78} \lor \neg \left(z \leq 1.65 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \cos y, x \cdot \frac{\sin y}{x}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\cos y}{z}, \frac{\sin y \cdot z}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 92.4

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

5. Applied rewrites 92.4%

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

6. Taylor expanded in z around inf

   \[\leadsto \left(z \cdot \left(\frac{\cos y}{z} + \frac{\sin y}{x}\right)\right) \cdot x \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 92.2

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

8. Applied rewrites 92.2%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))