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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 13

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

real(8) function code(x, y, z)
    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

real(8) function code(x, y, z)
    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

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

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

real(8) function code(x, y, z)
    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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -4000000000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -4000000000.0)
     (+ z x)
     (if (<= t_0 -0.05)
       (sin y)
       (if (<= t_0 0.002)
         (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))
         (if (<= t_0 1.0) (sin y) (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if (t_0 <= -4000000000.0) {
		tmp = z + x;
	} else if (t_0 <= -0.05) {
		tmp = sin(y);
	} else if (t_0 <= 0.002) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if (t_0 <= -4000000000.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -0.05)
		tmp = sin(y);
	elseif (t_0 <= 0.002)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4000000000.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 0.002], 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$0, 1.0], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -4e9 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 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 79.2

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

   5. Applied rewrites 79.2%

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

   if -4e9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.050000000000000003 or 2e-3 < (+.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

      2. lift--.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}}{-\cos y} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right)} - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \color{blue}{\left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}}{-\cos y} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \color{blue}{\left(1 \cdot \cos y\right)} \cdot \left(1 \cdot \cos y\right)}{-\cos y} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \color{blue}{\left(1 \cdot \cos y\right)}}{-\cos y} \]

      7. *-lft-identity N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \color{blue}{\cos y} \cdot \left(1 \cdot \cos y\right)}{-\cos y} \]

      8. *-lft-identity N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \cos y \cdot \color{blue}{\cos y}}{-\cos y} \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \cos y \cdot \cos y}{\color{blue}{\mathsf{neg}\left(\cos y\right)}} \]

      10. sub0-neg N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \cos y \cdot \cos y}{\color{blue}{0 - \cos y}} \]

      11. mul0-lft N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \cos y \cdot \cos y}{\color{blue}{0 \cdot \sin y} - \cos y} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \cos y \cdot \cos y}{\color{blue}{0 \cdot \sin y} - \cos y} \]

      13. flip-+ N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\left(0 \cdot \sin y + \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \left(\color{blue}{0 \cdot \sin y} + \cos y\right) \]

      15. mul0-lft N/A

          \[\leadsto \left(x + \sin y\right) + z \cdot \left(\color{blue}{0} + \cos y\right) \]

      16. +-lft-identity 100.0

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

      17. unpow1 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-pow1 N/A

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

      20. pow2 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 97.9%

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

   10. Taylor expanded in z around 0

       \[\leadsto \sin y \]
   11. Step-by-step derivation

   12. Applied rewrites 97.9%

       \[\leadsto \sin y \]

   if -0.050000000000000003 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2e-3

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 3: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.21 \lor \neg \left(t\_0 \leq 0.2\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (or (<= t_0 -0.21) (not (<= t_0 0.2))) (+ z x) (+ (+ z y) x))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if ((t_0 <= -0.21) || !(t_0 <= 0.2)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if ((t_0 <= (-0.21d0)) .or. (.not. (t_0 <= 0.2d0))) 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 t_0 = (x + Math.sin(y)) + (z * Math.cos(y));
	double tmp;
	if ((t_0 <= -0.21) || !(t_0 <= 0.2)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if (t_0 <= -0.21) or not (t_0 <= 0.2):
		tmp = z + x
	else:
		tmp = (z + y) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if ((t_0 <= -0.21) || !(t_0 <= 0.2))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(z + y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (z * cos(y));
	tmp = 0.0;
	if ((t_0 <= -0.21) || ~((t_0 <= 0.2)))
		tmp = z + x;
	else
		tmp = (z + y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.21], N[Not[LessEqual[t$95$0, 0.2]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.209999999999999992 or 0.20000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 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 69.7

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

   5. Applied rewrites 69.7%

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

   if -0.209999999999999992 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.20000000000000001

   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.21 \lor \neg \left(\left(x + \sin y\right) + z \cdot \cos y \leq 0.2\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 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. Add Preprocessing

Alternative 5: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-188}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -2.4e+95)
     t_0
     (if (<= z -1e-188) (+ z x) (if (<= z 3.4e+40) (+ (sin y) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -2.4e+95) {
		tmp = t_0;
	} else if (z <= -1e-188) {
		tmp = z + x;
	} else if (z <= 3.4e+40) {
		tmp = sin(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * z
    if (z <= (-2.4d+95)) then
        tmp = t_0
    else if (z <= (-1d-188)) then
        tmp = z + x
    else if (z <= 3.4d+40) then
        tmp = sin(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -2.4e+95) {
		tmp = t_0;
	} else if (z <= -1e-188) {
		tmp = z + x;
	} else if (z <= 3.4e+40) {
		tmp = Math.sin(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -2.4e+95:
		tmp = t_0
	elif z <= -1e-188:
		tmp = z + x
	elif z <= 3.4e+40:
		tmp = math.sin(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -2.4e+95)
		tmp = t_0;
	elseif (z <= -1e-188)
		tmp = Float64(z + x);
	elseif (z <= 3.4e+40)
		tmp = Float64(sin(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -2.4e+95)
		tmp = t_0;
	elseif (z <= -1e-188)
		tmp = z + x;
	elseif (z <= 3.4e+40)
		tmp = sin(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.4e+95], t$95$0, If[LessEqual[z, -1e-188], N[(z + x), $MachinePrecision], If[LessEqual[z, 3.4e+40], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e95 or 3.39999999999999989e40 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 84.2

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

   11. Applied rewrites 84.2%

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

   if -2.4e95 < z < -9.9999999999999995e-189

   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 81.8

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

   5. Applied rewrites 81.8%

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

   if -9.9999999999999995e-189 < z < 3.39999999999999989e40

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 100.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 96.1

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

   11. Applied rewrites 96.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+95}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-188}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+88}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\left(x + \sin y\right) + z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+88)
   (+ (+ y x) (* z (cos y)))
   (if (<= z 6e+142) (+ (+ x (sin y)) (* z 1.0)) (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+88) {
		tmp = (y + x) + (z * cos(y));
	} else if (z <= 6e+142) {
		tmp = (x + sin(y)) + (z * 1.0);
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.8d+88)) then
        tmp = (y + x) + (z * cos(y))
    else if (z <= 6d+142) then
        tmp = (x + sin(y)) + (z * 1.0d0)
    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 (z <= -5.8e+88) {
		tmp = (y + x) + (z * Math.cos(y));
	} else if (z <= 6e+142) {
		tmp = (x + Math.sin(y)) + (z * 1.0);
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.8e+88:
		tmp = (y + x) + (z * math.cos(y))
	elif z <= 6e+142:
		tmp = (x + math.sin(y)) + (z * 1.0)
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+88)
		tmp = Float64(Float64(y + x) + Float64(z * cos(y)));
	elseif (z <= 6e+142)
		tmp = Float64(Float64(x + sin(y)) + Float64(z * 1.0));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.8e+88)
		tmp = (y + x) + (z * cos(y));
	elseif (z <= 6e+142)
		tmp = (x + sin(y)) + (z * 1.0);
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.8e+88], N[(N[(y + x), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e+142], N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7999999999999999e88

   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 88.3

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

   5. Applied rewrites 88.3%

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

   if -5.7999999999999999e88 < z < 5.99999999999999949e142

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.2%

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

   if 5.99999999999999949e142 < z

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+88}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\left(x + \sin y\right) + z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-16}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e-16)
   (+ (+ y x) (* z (cos y)))
   (if (<= z 3.4e+40) (+ (sin y) x) (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e-16) {
		tmp = (y + x) + (z * cos(y));
	} else if (z <= 3.4e+40) {
		tmp = sin(y) + x;
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.4d-16)) then
        tmp = (y + x) + (z * cos(y))
    else if (z <= 3.4d+40) then
        tmp = sin(y) + 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 (z <= -5.4e-16) {
		tmp = (y + x) + (z * Math.cos(y));
	} else if (z <= 3.4e+40) {
		tmp = Math.sin(y) + x;
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.4e-16:
		tmp = (y + x) + (z * math.cos(y))
	elif z <= 3.4e+40:
		tmp = math.sin(y) + x
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e-16)
		tmp = Float64(Float64(y + x) + Float64(z * cos(y)));
	elseif (z <= 3.4e+40)
		tmp = Float64(sin(y) + x);
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.4e-16)
		tmp = (y + x) + (z * cos(y));
	elseif (z <= 3.4e+40)
		tmp = sin(y) + x;
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e-16], N[(N[(y + x), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e+40], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.39999999999999999e-16

   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -5.39999999999999999e-16 < z < 3.39999999999999989e40

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 100.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 93.2

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

   11. Applied rewrites 93.2%

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

   if 3.39999999999999989e40 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 87.4

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

   11. Applied rewrites 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-16}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.042 \lor \neg \left(y \leq 0.2\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.042) (not (<= y 0.2)))
   (+ (sin y) 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 <= -0.042) || !(y <= 0.2)) {
		tmp = sin(y) + 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 <= -0.042) || !(y <= 0.2))
		tmp = Float64(sin(y) + 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, -0.042], N[Not[LessEqual[y, 0.2]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] + 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 < -0.0420000000000000026 or 0.20000000000000001 < y

   1. Initial program 99.9%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. remove-double-neg N/A

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

      6. sin-neg-rev N/A

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

      7. cos-+PI/2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. cos-sum N/A

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

      11. cos-+PI/2-rev N/A

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

      12. sin-+PI/2-rev N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{\left(0 \cdot \sin y\right) \cdot \left(0 \cdot \sin y\right) - \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right)}{-\cos y}} \]

   6. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 66.9

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

   11. Applied rewrites 66.9%

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

   if -0.0420000000000000026 < y < 0.20000000000000001

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.042 \lor \neg \left(y \leq 0.2\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13500000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -13500000000.0) (not (<= y 3.6e-20)))
   (+ 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 <= -13500000000.0) || !(y <= 3.6e-20)) {
		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 <= -13500000000.0) || !(y <= 3.6e-20))
		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, -13500000000.0], N[Not[LessEqual[y, 3.6e-20]], $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 < -1.35e10 or 3.59999999999999974e-20 < 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 48.0

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

   5. Applied rewrites 48.0%

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

   if -1.35e10 < y < 3.59999999999999974e-20

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13500000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -25000000000.0) (not (<= y 3.6e-20)))
   (+ z x)
   (fma (fma (* y y) -0.5 1.0) z (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -25000000000.0) || !(y <= 3.6e-20)) {
		tmp = z + x;
	} else {
		tmp = fma(fma((y * y), -0.5, 1.0), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -25000000000.0) || !(y <= 3.6e-20))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(Float64(y * y), -0.5, 1.0), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -25000000000.0], N[Not[LessEqual[y, 3.6e-20]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e10 or 3.59999999999999974e-20 < 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 48.0

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

   5. Applied rewrites 48.0%

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

   if -2.5e10 < y < 3.59999999999999974e-20

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(y \cdot \left(\frac{1}{2} \cdot z - \frac{1}{y}\right)\right), y, z + x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-20}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-5} \lor \neg \left(x \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.7e-5) (not (<= x 1.75e-64))) (+ x y) (+ y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-5) || !(x <= 1.75e-64)) {
		tmp = x + y;
	} else {
		tmp = y + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.7d-5)) .or. (.not. (x <= 1.75d-64))) then
        tmp = x + y
    else
        tmp = y + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-5) || !(x <= 1.75e-64)) {
		tmp = x + y;
	} else {
		tmp = y + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.7e-5) or not (x <= 1.75e-64):
		tmp = x + y
	else:
		tmp = y + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.7e-5) || !(x <= 1.75e-64))
		tmp = Float64(x + y);
	else
		tmp = Float64(y + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.7e-5) || ~((x <= 1.75e-64)))
		tmp = x + y;
	else
		tmp = y + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.7e-5], N[Not[LessEqual[x, 1.75e-64]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e-5 or 1.7500000000000002e-64 < 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

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 14.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.4%

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

   if -1.7e-5 < x < 1.7500000000000002e-64

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.9%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-5} \lor \neg \left(x \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y + z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 68.3

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

5. Applied rewrites 68.3%

   \[\leadsto \color{blue}{z + x} \]
6. Add Preprocessing

Alternative 13: 38.5% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
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 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.0%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 40.8%

    \[\leadsto x + \color{blue}{y} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(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))))