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

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. log1p-expm1-u 99.7%

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

3. Applied egg-rr 99.7%

   \[\leadsto x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos y\right)\right)} - z \cdot \sin y \]
4. Step-by-step derivation

   1. log1p-expm1-u 99.8%

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

   2. +-rgt-identity 99.8%

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

   3. metadata-eval 99.8%

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

   4. associate--l+ 99.6%

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

   5. flip-- 99.6%

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

   6. metadata-eval 99.6%

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

   7. clear-num 99.6%

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

   8. un-div-inv 99.6%

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

   9. clear-num 99.6%

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

   10. metadata-eval 99.6%

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

   11. flip-- 99.6%

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

   12. associate--l+ 99.7%

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

   13. metadata-eval 99.7%

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

   14. +-rgt-identity 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-/r/ 99.8%

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

   2. fma-neg 99.8%

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

   3. /-rgt-identity 99.8%

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

   4. fma-def 99.8%

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

   5. *-commutative 99.8%

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

   6. fma-def 99.8%

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

   7. distribute-rgt-neg-in 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 86.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+70} \lor \neg \left(x \leq 2.8 \cdot 10^{+73}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.6e+70) (not (<= x 2.8e+73)))
   (* (cos y) x)
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+70) || !(x <= 2.8e+73)) {
		tmp = cos(y) * x;
	} else {
		tmp = x - (z * sin(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 <= (-4.6d+70)) .or. (.not. (x <= 2.8d+73))) then
        tmp = cos(y) * x
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+70) || !(x <= 2.8e+73)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.6e+70) or not (x <= 2.8e+73):
		tmp = math.cos(y) * x
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.6e+70) || !(x <= 2.8e+73))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.6e+70) || ~((x <= 2.8e+73)))
		tmp = cos(y) * x;
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.6e+70], N[Not[LessEqual[x, 2.8e+73]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.59999999999999987e70 or 2.80000000000000008e73 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 91.0%

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

   if -4.59999999999999987e70 < x < 2.80000000000000008e73

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 89.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+70} \lor \neg \left(x \leq 2.8 \cdot 10^{+73}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 4: 72.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+54} \lor \neg \left(x \leq 1.15 \cdot 10^{-64}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.4e+54) (not (<= x 1.15e-64)))
   (* (cos y) x)
   (* z (- (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e+54) || !(x <= 1.15e-64)) {
		tmp = cos(y) * x;
	} else {
		tmp = z * -sin(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 <= (-4.4d+54)) .or. (.not. (x <= 1.15d-64))) then
        tmp = cos(y) * x
    else
        tmp = z * -sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e+54) || !(x <= 1.15e-64)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = z * -Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.4e+54) or not (x <= 1.15e-64):
		tmp = math.cos(y) * x
	else:
		tmp = z * -math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.4e+54) || !(x <= 1.15e-64))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.4e+54) || ~((x <= 1.15e-64)))
		tmp = cos(y) * x;
	else
		tmp = z * -sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.4e+54], N[Not[LessEqual[x, 1.15e-64]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e54 or 1.1500000000000001e-64 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 84.2%

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

   if -4.3999999999999998e54 < x < 1.1500000000000001e-64

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 72.6%

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

      1. neg-mul-1 72.6%

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

      2. *-commutative 72.6%

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

      3. distribute-rgt-neg-in 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+54} \lor \neg \left(x \leq 1.15 \cdot 10^{-64}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.026 \lor \neg \left(y \leq 0.0185\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.026) (not (<= y 0.0185))) (* (cos y) x) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.026) || !(y <= 0.0185)) {
		tmp = cos(y) * x;
	} else {
		tmp = x - (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 ((y <= (-0.026d0)) .or. (.not. (y <= 0.0185d0))) then
        tmp = cos(y) * x
    else
        tmp = x - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.026) || !(y <= 0.0185)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.026) or not (y <= 0.0185):
		tmp = math.cos(y) * x
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.026) || !(y <= 0.0185))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.026) || ~((y <= 0.0185)))
		tmp = cos(y) * x;
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.026], N[Not[LessEqual[y, 0.0185]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0259999999999999988 or 0.0184999999999999991 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 52.0%

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

   if -0.0259999999999999988 < y < 0.0184999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.026 \lor \neg \left(y \leq 0.0185\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 6: 38.1% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. add-sqr-sqrt 55.1%

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

   4. associate-*r* 55.1%

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

   5. fma-def 55.1%

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

   6. distribute-rgt-neg-in 55.1%

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

   7. add-sqr-sqrt 28.3%

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

   8. sqrt-unprod 40.6%

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

   9. sqr-neg 40.6%

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

   10. sqrt-unprod 14.7%

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

   11. add-sqr-sqrt 29.6%

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

3. Applied egg-rr 29.6%

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

4. Taylor expanded in y around 0 31.3%

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

5. Final simplification 31.3%

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

Alternative 7: 52.2% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 46.5%

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

   1. mul-1-neg 46.5%

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

   2. unsub-neg 46.5%

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

4. Simplified 46.5%

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

5. Final simplification 46.5%

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

Alternative 8: 3.3% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. add-sqr-sqrt 55.1%

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

   4. associate-*r* 55.1%

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

   5. fma-def 55.1%

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

   6. distribute-rgt-neg-in 55.1%

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

   7. add-sqr-sqrt 28.3%

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

   8. sqrt-unprod 40.6%

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

   9. sqr-neg 40.6%

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

   10. sqrt-unprod 14.7%

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

   11. add-sqr-sqrt 29.6%

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

3. Applied egg-rr 29.6%

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

4. Taylor expanded in y around 0 31.3%

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

5. Taylor expanded in x around 0 3.4%

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

6. Final simplification 3.4%

   \[\leadsto y \cdot z \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))