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

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.0057:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{elif}\;y \leq 10^{+238}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.0057)
     t_0
     (if (<= y 0.0026)
       (fma y (fma z (* y -0.5) x) z)
       (if (<= y 1e+238) t_0 (* (cos y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.0057) {
		tmp = t_0;
	} else if (y <= 0.0026) {
		tmp = fma(y, fma(z, (y * -0.5), x), z);
	} else if (y <= 1e+238) {
		tmp = t_0;
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.0057)
		tmp = t_0;
	elseif (y <= 0.0026)
		tmp = fma(y, fma(z, Float64(y * -0.5), x), z);
	elseif (y <= 1e+238)
		tmp = t_0;
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0057], t$95$0, If[LessEqual[y, 0.0026], N[(y * N[(z * N[(y * -0.5), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[y, 1e+238], t$95$0, N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0057000000000000002 or 0.0025999999999999999 < y < 1e238

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \sin y + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, 0\right)} \]

      3. sin-lowering-sin.f64 56.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sin y}, 0\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 56.2

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

   7. Applied egg-rr 56.2%

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

   if -0.0057000000000000002 < y < 0.0025999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1e238 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 69.7

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

   7. Simplified 69.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0057:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{elif}\;y \leq 10^{+238}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) x z)))
   (if (<= x -4.4e-71) t_0 (if (<= x 1.45e-77) (* (cos y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), x, z);
	double tmp;
	if (x <= -4.4e-71) {
		tmp = t_0;
	} else if (x <= 1.45e-77) {
		tmp = cos(y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), x, z)
	tmp = 0.0
	if (x <= -4.4e-71)
		tmp = t_0;
	elseif (x <= 1.45e-77)
		tmp = Float64(cos(y) * z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision]}, If[LessEqual[x, -4.4e-71], t$95$0, If[LessEqual[x, 1.45e-77], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999995e-71 or 1.4499999999999999e-77 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 87.9%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sin-lowering-sin.f64 87.9

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

   7. Applied egg-rr 87.9%

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

   if -4.39999999999999995e-71 < x < 1.4499999999999999e-77

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 92.7

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

   7. Simplified 92.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;y \leq -43:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= y -43.0)
     t_0
     (if (<= y 0.5)
       (fma y (fma y (fma z -0.5 (* x (* y -0.16666666666666666))) x) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (y <= -43.0) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = fma(y, fma(y, fma(z, -0.5, (x * (y * -0.16666666666666666))), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (y <= -43.0)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(x * Float64(y * -0.16666666666666666))), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -43.0], t$95$0, If[LessEqual[y, 0.5], N[(y * N[(y * N[(z * -0.5 + N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -43 or 0.5 < y

   1. Initial program 99.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 48.2

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

   7. Simplified 48.2%

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

   if -43 < y < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -43:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+190}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+199}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+190) (* y x) (if (<= x 6.2e+199) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+190) {
		tmp = y * x;
	} else if (x <= 6.2e+199) {
		tmp = z;
	} else {
		tmp = 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) :: tmp
    if (x <= (-1.3d+190)) then
        tmp = y * x
    else if (x <= 6.2d+199) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+190) {
		tmp = y * x;
	} else if (x <= 6.2e+199) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+190:
		tmp = y * x
	elif x <= 6.2e+199:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e+190)
		tmp = Float64(y * x);
	elseif (x <= 6.2e+199)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e+190)
		tmp = y * x;
	elseif (x <= 6.2e+199)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+190], N[(y * x), $MachinePrecision], If[LessEqual[x, 6.2e+199], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000005e190 or 6.19999999999999971e199 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \sin y + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, 0\right)} \]

      3. sin-lowering-sin.f64 88.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sin y}, 0\right) \]

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, 0\right)} \]

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 45.7

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

   8. Simplified 45.7%

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

   if -1.30000000000000005e190 < x < 6.19999999999999971e199

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 40.9%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+190}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+199}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 51.8

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

5. Simplified 51.8%

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

Alternative 7: 38.8% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 35.1%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))