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

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 9

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Final simplification 99.8%

   \[\leadsto z \cdot \cos y + x \cdot \sin y \]

Alternative 3: 75.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ t_1 := z \cdot \cos y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.0022:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+78} \lor \neg \left(y \leq 2 \cdot 10^{+160}\right) \land y \leq 1.4 \cdot 10^{+274}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))) (t_1 (* z (cos y))))
   (if (<= y -2e+242)
     t_0
     (if (<= y -2.15e+162)
       t_1
       (if (<= y -2.2e+123)
         t_0
         (if (<= y -0.0022)
           t_1
           (if (<= y 7.5e-12)
             (+ (* -0.5 (* z (* y y))) (+ z (* x y)))
             (if (or (<= y 1.02e+78) (and (not (<= y 2e+160)) (<= y 1.4e+274)))
               t_1
               t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double t_1 = z * cos(y);
	double tmp;
	if (y <= -2e+242) {
		tmp = t_0;
	} else if (y <= -2.15e+162) {
		tmp = t_1;
	} else if (y <= -2.2e+123) {
		tmp = t_0;
	} else if (y <= -0.0022) {
		tmp = t_1;
	} else if (y <= 7.5e-12) {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	} else if ((y <= 1.02e+78) || (!(y <= 2e+160) && (y <= 1.4e+274))) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * sin(y)
    t_1 = z * cos(y)
    if (y <= (-2d+242)) then
        tmp = t_0
    else if (y <= (-2.15d+162)) then
        tmp = t_1
    else if (y <= (-2.2d+123)) then
        tmp = t_0
    else if (y <= (-0.0022d0)) then
        tmp = t_1
    else if (y <= 7.5d-12) then
        tmp = ((-0.5d0) * (z * (y * y))) + (z + (x * y))
    else if ((y <= 1.02d+78) .or. (.not. (y <= 2d+160)) .and. (y <= 1.4d+274)) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double t_1 = z * Math.cos(y);
	double tmp;
	if (y <= -2e+242) {
		tmp = t_0;
	} else if (y <= -2.15e+162) {
		tmp = t_1;
	} else if (y <= -2.2e+123) {
		tmp = t_0;
	} else if (y <= -0.0022) {
		tmp = t_1;
	} else if (y <= 7.5e-12) {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	} else if ((y <= 1.02e+78) || (!(y <= 2e+160) && (y <= 1.4e+274))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	t_1 = z * math.cos(y)
	tmp = 0
	if y <= -2e+242:
		tmp = t_0
	elif y <= -2.15e+162:
		tmp = t_1
	elif y <= -2.2e+123:
		tmp = t_0
	elif y <= -0.0022:
		tmp = t_1
	elif y <= 7.5e-12:
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y))
	elif (y <= 1.02e+78) or (not (y <= 2e+160) and (y <= 1.4e+274)):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	t_1 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -2e+242)
		tmp = t_0;
	elseif (y <= -2.15e+162)
		tmp = t_1;
	elseif (y <= -2.2e+123)
		tmp = t_0;
	elseif (y <= -0.0022)
		tmp = t_1;
	elseif (y <= 7.5e-12)
		tmp = Float64(Float64(-0.5 * Float64(z * Float64(y * y))) + Float64(z + Float64(x * y)));
	elseif ((y <= 1.02e+78) || (!(y <= 2e+160) && (y <= 1.4e+274)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	t_1 = z * cos(y);
	tmp = 0.0;
	if (y <= -2e+242)
		tmp = t_0;
	elseif (y <= -2.15e+162)
		tmp = t_1;
	elseif (y <= -2.2e+123)
		tmp = t_0;
	elseif (y <= -0.0022)
		tmp = t_1;
	elseif (y <= 7.5e-12)
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	elseif ((y <= 1.02e+78) || (~((y <= 2e+160)) && (y <= 1.4e+274)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+242], t$95$0, If[LessEqual[y, -2.15e+162], t$95$1, If[LessEqual[y, -2.2e+123], t$95$0, If[LessEqual[y, -0.0022], t$95$1, If[LessEqual[y, 7.5e-12], N[(N[(-0.5 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.02e+78], And[N[Not[LessEqual[y, 2e+160]], $MachinePrecision], LessEqual[y, 1.4e+274]]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000001e242 or -2.1500000000000001e162 < y < -2.19999999999999992e123 or 1.01999999999999994e78 < y < 2.00000000000000001e160 or 1.40000000000000004e274 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 59.3%

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

      4. associate-*r* 59.3%

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

      5. fma-def 59.3%

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

   3. Applied egg-rr 59.3%

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

      1. fma-udef 59.3%

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

      2. associate-*r* 59.3%

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

      3. add-sqr-sqrt 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      7. unpow2 98.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{x}\right) \cdot \sin y + z \cdot \cos y \]

      8. associate-*r* 98.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      9. fma-udef 98.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      10. add-sqr-sqrt 41.7%

          \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \cdot \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)}} \]

      11. pow2 41.7%

          \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)}\right)}^{2}} \]

   5. Applied egg-rr 42.0%

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

      1. unpow2 42.0%

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

      2. add-sqr-sqrt 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. add-cube-cbrt 99.2%

         \[\leadsto \cos y \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} + x \cdot \sin y \]

      6. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\cos y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} + x \cdot \sin y \]

      7. fma-def 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, x \cdot \sin y\right)} \]

      8. pow2 99.2%

         \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}, \sqrt[3]{z}, x \cdot \sin y\right) \]

      9. *-commutative 99.2%

         \[\leadsto \mathsf{fma}\left(\cos y \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, \color{blue}{\sin y \cdot x}\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, \sin y \cdot x\right)} \]

   8. Taylor expanded in z around 0 73.9%

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

   if -2.0000000000000001e242 < y < -2.1500000000000001e162 or -2.19999999999999992e123 < y < -0.00220000000000000013 or 7.5e-12 < y < 1.01999999999999994e78 or 2.00000000000000001e160 < y < 1.40000000000000004e274

   1. Initial program 99.6%

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

      1. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      2. associate-*l* 99.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      3. fma-def 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      4. pow2 99.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 73.2%

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

   if -0.00220000000000000013 < y < 7.5e-12

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. *-commutative 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{z \cdot {y}^{2}}\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. unpow2 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(z \cdot \color{blue}{\left(y \cdot y\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

      5. associate-*r* 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(z \cdot y\right) \cdot y}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 96.7%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+242}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -0.0022:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+78} \lor \neg \left(y \leq 2 \cdot 10^{+160}\right) \land y \leq 1.4 \cdot 10^{+274}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 4: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ t_1 := z \cdot \cos y\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+75} \lor \neg \left(y \leq 4.1 \cdot 10^{+164}\right) \land y \leq 2.8 \cdot 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))) (t_1 (* z (cos y))))
   (if (<= y -2.7e+242)
     t_0
     (if (<= y -2.35e+162)
       t_1
       (if (<= y -1.75e+118)
         t_0
         (if (<= y -2.1e-8)
           t_1
           (if (<= y 7.5e-12)
             (fma y x z)
             (if (or (<= y 3.55e+75)
                     (and (not (<= y 4.1e+164)) (<= y 2.8e+273)))
               t_1
               t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double t_1 = z * cos(y);
	double tmp;
	if (y <= -2.7e+242) {
		tmp = t_0;
	} else if (y <= -2.35e+162) {
		tmp = t_1;
	} else if (y <= -1.75e+118) {
		tmp = t_0;
	} else if (y <= -2.1e-8) {
		tmp = t_1;
	} else if (y <= 7.5e-12) {
		tmp = fma(y, x, z);
	} else if ((y <= 3.55e+75) || (!(y <= 4.1e+164) && (y <= 2.8e+273))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	t_1 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -2.7e+242)
		tmp = t_0;
	elseif (y <= -2.35e+162)
		tmp = t_1;
	elseif (y <= -1.75e+118)
		tmp = t_0;
	elseif (y <= -2.1e-8)
		tmp = t_1;
	elseif (y <= 7.5e-12)
		tmp = fma(y, x, z);
	elseif ((y <= 3.55e+75) || (!(y <= 4.1e+164) && (y <= 2.8e+273)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e+242], t$95$0, If[LessEqual[y, -2.35e+162], t$95$1, If[LessEqual[y, -1.75e+118], t$95$0, If[LessEqual[y, -2.1e-8], t$95$1, If[LessEqual[y, 7.5e-12], N[(y * x + z), $MachinePrecision], If[Or[LessEqual[y, 3.55e+75], And[N[Not[LessEqual[y, 4.1e+164]], $MachinePrecision], LessEqual[y, 2.8e+273]]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.69999999999999984e242 or -2.35000000000000001e162 < y < -1.75000000000000008e118 or 3.54999999999999991e75 < y < 4.10000000000000016e164 or 2.80000000000000018e273 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 59.3%

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

      4. associate-*r* 59.3%

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

      5. fma-def 59.3%

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

   3. Applied egg-rr 59.3%

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

      1. fma-udef 59.3%

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

      2. associate-*r* 59.3%

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

      3. add-sqr-sqrt 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      7. unpow2 98.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{x}\right) \cdot \sin y + z \cdot \cos y \]

      8. associate-*r* 98.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      9. fma-udef 98.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      10. add-sqr-sqrt 41.7%

          \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \cdot \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)}} \]

      11. pow2 41.7%

          \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)}\right)}^{2}} \]

   5. Applied egg-rr 42.0%

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

      1. unpow2 42.0%

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

      2. add-sqr-sqrt 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. add-cube-cbrt 99.2%

         \[\leadsto \cos y \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} + x \cdot \sin y \]

      6. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\cos y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} + x \cdot \sin y \]

      7. fma-def 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, x \cdot \sin y\right)} \]

      8. pow2 99.2%

         \[\leadsto \mathsf{fma}\left(\cos y \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}, \sqrt[3]{z}, x \cdot \sin y\right) \]

      9. *-commutative 99.2%

         \[\leadsto \mathsf{fma}\left(\cos y \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, \color{blue}{\sin y \cdot x}\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, \sin y \cdot x\right)} \]

   8. Taylor expanded in z around 0 73.9%

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

   if -2.69999999999999984e242 < y < -2.35000000000000001e162 or -1.75000000000000008e118 < y < -2.09999999999999994e-8 or 7.5e-12 < y < 3.54999999999999991e75 or 4.10000000000000016e164 < y < 2.80000000000000018e273

   1. Initial program 99.6%

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

      1. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      2. associate-*l* 99.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      3. fma-def 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      4. pow2 99.3%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 73.8%

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

   if -2.09999999999999994e-8 < y < 7.5e-12

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 100.0%

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

      1. fma-def 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+242}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+162}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+75} \lor \neg \left(y \leq 4.1 \cdot 10^{+164}\right) \land y \leq 2.8 \cdot 10^{+273}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]

Alternative 5: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-72} \lor \neg \left(x \leq 1.32 \cdot 10^{-33}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.12e-72) (not (<= x 1.32e-33)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e-72) || !(x <= 1.32e-33)) {
		tmp = z + (x * sin(y));
	} else {
		tmp = z * cos(y);
	}
	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 <= (-1.12d-72)) .or. (.not. (x <= 1.32d-33))) then
        tmp = z + (x * sin(y))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e-72) || !(x <= 1.32e-33)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.12e-72) or not (x <= 1.32e-33):
		tmp = z + (x * math.sin(y))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.12e-72) || !(x <= 1.32e-33))
		tmp = Float64(z + Float64(x * sin(y)));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.12e-72) || ~((x <= 1.32e-33)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.12e-72], N[Not[LessEqual[x, 1.32e-33]], $MachinePrecision]], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12000000000000005e-72 or 1.31999999999999993e-33 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 88.6%

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

   if -1.12000000000000005e-72 < x < 1.31999999999999993e-33

   1. Initial program 99.8%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      2. associate-*l* 99.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      3. fma-def 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      4. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right) \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 90.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-72} \lor \neg \left(x \leq 1.32 \cdot 10^{-33}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 6: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0026 \lor \neg \left(y \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0026) (not (<= y 7.5e-12)))
   (* z (cos y))
   (+ (* -0.5 (* z (* y y))) (+ z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0026) || !(y <= 7.5e-12)) {
		tmp = z * cos(y);
	} else {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	}
	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 ((y <= (-0.0026d0)) .or. (.not. (y <= 7.5d-12))) then
        tmp = z * cos(y)
    else
        tmp = ((-0.5d0) * (z * (y * y))) + (z + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0026) || !(y <= 7.5e-12)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0026) or not (y <= 7.5e-12):
		tmp = z * math.cos(y)
	else:
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0026) || !(y <= 7.5e-12))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(-0.5 * Float64(z * Float64(y * y))) + Float64(z + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0026) || ~((y <= 7.5e-12)))
		tmp = z * cos(y);
	else
		tmp = (-0.5 * (z * (y * y))) + (z + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0026], N[Not[LessEqual[y, 7.5e-12]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0025999999999999999 or 7.5e-12 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sin y + z \cdot \cos y \]

      2. associate-*l* 98.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sin y\right)} + z \cdot \cos y \]

      3. fma-def 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

      4. pow2 98.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right) \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x} \cdot \sin y, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 55.1%

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

   if -0.0025999999999999999 < y < 7.5e-12

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. *-commutative 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{z \cdot {y}^{2}}\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. unpow2 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(z \cdot \color{blue}{\left(y \cdot y\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

      5. associate-*r* 96.7%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(z \cdot y\right) \cdot y}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 96.7%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 96.7%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0026 \lor \neg \left(y \leq 7.5 \cdot 10^{-12}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \]

Alternative 7: 43.1% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-154}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-191}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.26e-154) z (if (<= z 1.5e-191) (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.26e-154) {
		tmp = z;
	} else if (z <= 1.5e-191) {
		tmp = x * y;
	} else {
		tmp = 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 <= (-1.26d-154)) then
        tmp = z
    else if (z <= 1.5d-191) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.26e-154) {
		tmp = z;
	} else if (z <= 1.5e-191) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.26e-154:
		tmp = z
	elif z <= 1.5e-191:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.26e-154)
		tmp = z;
	elseif (z <= 1.5e-191)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.26e-154)
		tmp = z;
	elseif (z <= 1.5e-191)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.26e-154], z, If[LessEqual[z, 1.5e-191], N[(x * y), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25999999999999994e-154 or 1.5e-191 < z

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 49.4%

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

   3. Taylor expanded in y around 0 41.5%

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

   if -1.25999999999999994e-154 < z < 1.5e-191

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 58.4%

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

   3. Taylor expanded in y around inf 45.0%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-154}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-191}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 8: 53.4% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Taylor expanded in y around 0 51.0%

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

3. Final simplification 51.0%

   \[\leadsto z + x \cdot y \]

Alternative 9: 39.6% accurate, 207.0× 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. Taylor expanded in y around 0 51.0%

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

3. Taylor expanded in y around 0 36.7%

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

4. Final simplification 36.7%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023173 
(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))))