Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 9.9s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

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: 99.9% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.0× speedup

Math

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

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 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-1}{x}, \left(\cos y \cdot \left(-x\right)\right) \cdot z, x\right)\\ t_1 := \left(x + \sin y\right) + z \cdot \cos y\\ t_2 := \mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ -1.0 x) (* (* (cos y) (- x)) z) x))
        (t_1 (+ (+ x (sin y)) (* z (cos y))))
        (t_2 (fma (cos y) z (+ x y))))
   (if (<= t_1 -5e+229)
     t_2
     (if (<= t_1 -5000000000.0)
       t_0
       (if (<= t_1 50.0)
         (fma 1.0 z (+ (sin y) x))
         (if (<= t_1 1e+154) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-1.0 / x), ((cos(y) * -x) * z), x);
	double t_1 = (x + sin(y)) + (z * cos(y));
	double t_2 = fma(cos(y), z, (x + y));
	double tmp;
	if (t_1 <= -5e+229) {
		tmp = t_2;
	} else if (t_1 <= -5000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 50.0) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else if (t_1 <= 1e+154) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-1.0 / x), Float64(Float64(cos(y) * Float64(-x)) * z), x)
	t_1 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	t_2 = fma(cos(y), z, Float64(x + y))
	tmp = 0.0
	if (t_1 <= -5e+229)
		tmp = t_2;
	elseif (t_1 <= -5000000000.0)
		tmp = t_0;
	elseif (t_1 <= 50.0)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	elseif (t_1 <= 1e+154)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * (-x)), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+229], t$95$2, If[LessEqual[t$95$1, -5000000000.0], t$95$0, If[LessEqual[t$95$1, 50.0], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+154], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5.0000000000000005e229 or 1.00000000000000004e154 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 90.8

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

   5. Applied rewrites 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -5.0000000000000005e229 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e9 or 50 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1.00000000000000004e154

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. rgt-mult-inverse N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

      11. rgt-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 88.1%

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 96.8%

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

   if -5e9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 50

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \sin y + x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -5 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq -5000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x}, \left(\cos y \cdot \left(-x\right)\right) \cdot z, x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 50:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x}, \left(\cos y \cdot \left(-x\right)\right) \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1, z, \sin y\right)\\ t_1 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 1.0 z (sin y))) (t_1 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_1 -50000.0)
     (+ z x)
     (if (<= t_1 -0.05)
       t_0
       (if (<= t_1 5e-24)
         (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))
         (if (<= t_1 1.0) t_0 (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(1.0, z, sin(y));
	double t_1 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = z + x;
	} else if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 5e-24) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(1.0, z, sin(y))
	t_1 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = Float64(z + x);
	elseif (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 5e-24)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 * z + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 5e-24], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], t$95$0, N[(z + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e4 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -5e4 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.050000000000000003 or 4.9999999999999998e-24 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(1, z, \sin y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.8%

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

   if -0.050000000000000003 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 4.9999999999999998e-24

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -50000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+146} \lor \neg \left(z \leq 1.7 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e+146) (not (<= z 1.7e+21)))
   (fma (cos y) z (+ x y))
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+146) || !(z <= 1.7e+21)) {
		tmp = fma(cos(y), z, (x + y));
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e+146) || !(z <= 1.7e+21))
		tmp = fma(cos(y), z, Float64(x + y));
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e+146], N[Not[LessEqual[z, 1.7e+21]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000001e146 or 1.7e21 < z

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.7

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

   5. Applied rewrites 84.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -1.50000000000000001e146 < z < 1.7e21

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 92.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \sin y + x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+146} \lor \neg \left(z \leq 1.7 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+93} \lor \neg \left(z \leq 9.8 \cdot 10^{+36}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.1e+93) (not (<= z 9.8e+36)))
   (* (cos y) z)
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.1e+93) || !(z <= 9.8e+36)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.1e+93) || !(z <= 9.8e+36))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.1e+93], N[Not[LessEqual[z, 9.8e+36]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000001e93 or 9.79999999999999962e36 < z

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. rgt-mult-inverse N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

      11. rgt-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 78.0%

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

   9. Applied rewrites 66.6%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 77.0%

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

   if -4.1000000000000001e93 < z < 9.79999999999999962e36

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 94.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \sin y + x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+93} \lor \neg \left(z \leq 9.8 \cdot 10^{+36}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000 \lor \neg \left(x \leq 3.5 \cdot 10^{-81}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2000.0) (not (<= x 3.5e-81))) (+ z x) (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2000.0) || !(x <= 3.5e-81)) {
		tmp = z + x;
	} else {
		tmp = cos(y) * 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 <= (-2000.0d0)) .or. (.not. (x <= 3.5d-81))) then
        tmp = z + x
    else
        tmp = cos(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2000.0) || !(x <= 3.5e-81)) {
		tmp = z + x;
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2000.0) or not (x <= 3.5e-81):
		tmp = z + x
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2000.0) || !(x <= 3.5e-81))
		tmp = Float64(z + x);
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2000.0) || ~((x <= 3.5e-81)))
		tmp = z + x;
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2000.0], N[Not[LessEqual[x, 3.5e-81]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e3 or 3.49999999999999986e-81 < x

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if -2e3 < x < 3.49999999999999986e-81

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. rgt-mult-inverse N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

      11. rgt-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 93.6%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 58.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000 \lor \neg \left(x \leq 3.5 \cdot 10^{-81}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+18} \lor \neg \left(y \leq 54000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5 \cdot z\right), y \cdot y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7e+18) (not (<= y 54000.0)))
   (+ z x)
   (+
    (+ y x)
    (fma
     (fma
      (* z (fma -0.001388888888888889 (* y y) 0.041666666666666664))
      (* y y)
      (* -0.5 z))
     (* y y)
     z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e+18) || !(y <= 54000.0)) {
		tmp = z + x;
	} else {
		tmp = (y + x) + fma(fma((z * fma(-0.001388888888888889, (y * y), 0.041666666666666664)), (y * y), (-0.5 * z)), (y * y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.7e+18) || !(y <= 54000.0))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(y + x) + fma(fma(Float64(z * fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664)), Float64(y * y), Float64(-0.5 * z)), Float64(y * y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.7e+18], N[Not[LessEqual[y, 54000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] + N[(N[(N[(z * N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e18 or 54000 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if -1.7e18 < y < 54000

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right) + z\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y + x\right) + \left(\color{blue}{\left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right)\right) \cdot {y}^{2}} + z\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{1}{24} \cdot \left({y}^{2} \cdot z\right), {y}^{2}, z\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{2} \cdot z}, {y}^{2}, z\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{2} \cdot z, {y}^{2}, z\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right)}, {y}^{2}, z\right) \]

      7. *-commutative N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right) \cdot z}, {y}^{2}, z\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right) \cdot z}, {y}^{2}, z\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {y}^{2}, \frac{-1}{2}\right)} \cdot z, {y}^{2}, z\right) \]

      10. unpow2 N/A

          \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right) \cdot z, {y}^{2}, z\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right) \cdot z, {y}^{2}, z\right) \]

      12. unpow2 N/A

          \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, y \cdot y, \frac{-1}{2}\right) \cdot z, \color{blue}{y \cdot y}, z\right) \]

      13. lower-*.f64 97.8

          \[\leadsto \left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right) \cdot z, \color{blue}{y \cdot y}, z\right) \]

   8. Applied rewrites 97.8%

      \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right) \cdot z, y \cdot y, z\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(y + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(y + x\right) + \left(\color{blue}{\left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) \cdot {y}^{2}} + z\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right), {y}^{2}, z\right)} \]

   11. Applied rewrites 98.0%

       \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5 \cdot z\right), y \cdot y, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+18} \lor \neg \left(y \leq 54000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5 \cdot z\right), y \cdot y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000000000 \lor \neg \left(y \leq 110000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -16000000000.0) (not (<= y 110000.0)))
   (+ z x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -16000000000.0) || !(y <= 110000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -16000000000.0) || !(y <= 110000.0))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -16000000000.0], N[Not[LessEqual[y, 110000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e10 or 1.1e5 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 46.7

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

   5. Applied rewrites 46.7%

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

   if -1.6e10 < y < 1.1e5

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 98.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000000000 \lor \neg \left(y \leq 110000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+18} \lor \neg \left(y \leq 110000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, z \cdot y, 1\right), y, x\right) + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.4e+18) (not (<= y 110000.0)))
   (+ z x)
   (+ (fma (fma -0.5 (* z y) 1.0) y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.4e+18) || !(y <= 110000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(-0.5, (z * y), 1.0), y, x) + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.4e+18) || !(y <= 110000.0))
		tmp = Float64(z + x);
	else
		tmp = Float64(fma(fma(-0.5, Float64(z * y), 1.0), y, x) + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.4e+18], N[Not[LessEqual[y, 110000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.5 * N[(z * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + x), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4e18 or 1.1e5 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if -6.4e18 < y < 1.1e5

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, \color{blue}{z + x}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, z \cdot y, 1\right), y, x\right) + \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+18} \lor \neg \left(y \leq 110000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, z \cdot y, 1\right), y, x\right) + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+18} \lor \neg \left(y \leq 70000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7e+18) (not (<= y 70000.0))) (+ z x) (+ (+ z y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e+18) || !(y <= 70000.0)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	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 ((y <= (-1.7d+18)) .or. (.not. (y <= 70000.0d0))) then
        tmp = z + x
    else
        tmp = (z + y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e+18) || !(y <= 70000.0)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.7e+18) or not (y <= 70000.0):
		tmp = z + x
	else:
		tmp = (z + y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.7e+18) || !(y <= 70000.0))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(z + y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.7e+18) || ~((y <= 70000.0)))
		tmp = z + x;
	else
		tmp = (z + y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.7e+18], N[Not[LessEqual[y, 70000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e18 or 7e4 < y

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if -1.7e18 < y < 7e4

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 97.3

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

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\left(z + y\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+18} \lor \neg \left(y \leq 70000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-189} \lor \neg \left(x \leq 8.2 \cdot 10^{-201}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-189) (not (<= x 8.2e-201))) (+ z x) (+ z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-189) || !(x <= 8.2e-201)) {
		tmp = z + x;
	} else {
		tmp = z + y;
	}
	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 <= (-1.1d-189)) .or. (.not. (x <= 8.2d-201))) then
        tmp = z + x
    else
        tmp = z + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-189) || !(x <= 8.2e-201)) {
		tmp = z + x;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-189) or not (x <= 8.2e-201):
		tmp = z + x
	else:
		tmp = z + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e-189) || !(x <= 8.2e-201))
		tmp = Float64(z + x);
	else
		tmp = Float64(z + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.1e-189) || ~((x <= 8.2e-201)))
		tmp = z + x;
	else
		tmp = z + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-189], N[Not[LessEqual[x, 8.2e-201]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-189 or 8.20000000000000003e-201 < x

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -1.1000000000000001e-189 < x < 8.20000000000000003e-201

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-189} \lor \neg \left(x \leq 8.2 \cdot 10^{-201}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.4% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ z + y \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.5%

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

9. Final simplification 26.5%

   \[\leadsto z + y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025016 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))