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

Percentage Accurate: 99.8% → 99.8%

Time: 12.3s

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 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+210}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.015:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -3.8e+210)
     t_0
     (if (<= y -0.015)
       (* (cos y) z)
       (if (<= y 0.035) (fma y (fma z (* y -0.5) x) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -3.8e+210) {
		tmp = t_0;
	} else if (y <= -0.015) {
		tmp = cos(y) * z;
	} else if (y <= 0.035) {
		tmp = fma(y, fma(z, (y * -0.5), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -3.8e+210)
		tmp = t_0;
	elseif (y <= -0.015)
		tmp = Float64(cos(y) * z);
	elseif (y <= 0.035)
		tmp = fma(y, fma(z, Float64(y * -0.5), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e+210], t$95$0, If[LessEqual[y, -0.015], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 0.035], N[(y * N[(z * N[(y * -0.5), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.80000000000000028e210 or 0.035000000000000003 < y

   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. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 56.0

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

   5. Simplified 56.0%

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

   if -3.80000000000000028e210 < y < -0.014999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. cos-lowering-cos.f64 57.1

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

   5. Simplified 57.1%

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

   if -0.014999999999999999 < y < 0.035000000000000003

   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%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+210}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -0.015:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) x z)))
   (if (<= x -4.8e-28) t_0 (if (<= x 2.2e-152) (* (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 <= -4.8e-28) {
		tmp = t_0;
	} else if (x <= 2.2e-152) {
		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 <= -4.8e-28)
		tmp = t_0;
	elseif (x <= 2.2e-152)
		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, -4.8e-28], t$95$0, If[LessEqual[x, 2.2e-152], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000004e-28 or 2.19999999999999985e-152 < x

   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 99.9

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

   4. Applied egg-rr 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. cos-lowering-cos.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 88.2%

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

   if -4.8000000000000004e-28 < x < 2.19999999999999985e-152

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. cos-lowering-cos.f64 93.7

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

   5. Simplified 93.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0054:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z \cdot -0.5\right), y, \mathsf{fma}\left(y, x, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -52000000.0)
     t_0
     (if (<= y 0.0054) (fma (* y (* z -0.5)) y (fma y x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -52000000.0) {
		tmp = t_0;
	} else if (y <= 0.0054) {
		tmp = fma((y * (z * -0.5)), y, fma(y, x, z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -52000000.0)
		tmp = t_0;
	elseif (y <= 0.0054)
		tmp = fma(Float64(y * Float64(z * -0.5)), y, fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -52000000.0], t$95$0, If[LessEqual[y, 0.0054], N[(N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision] * y + N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e7 or 0.0054000000000000003 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sin-lowering-sin.f64 53.8

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

   5. Simplified 53.8%

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

   if -5.2e7 < y < 0.0054000000000000003

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 97.6

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

   7. Simplified 97.6%

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

      1. remove-double-div N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 97.8

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

   9. Applied egg-rr 97.8%

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

Alternative 5: 40.2% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+191}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 4.5e+191) z (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+191) {
		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 <= 4.5d+191) 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 <= 4.5e+191) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.5e+191:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e+191)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.5e+191)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e+191], z, N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000002e191

   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 43.1%

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

   if 4.5000000000000002e191 < 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. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 75.2

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

   5. Simplified 75.2%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 34.8

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

   8. Simplified 34.8%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+191}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% 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 51.2

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

5. Simplified 51.2%

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

Alternative 7: 39.2% 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 40.3%

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

Reproduce

herbie shell --seed 2024198 
(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))))