Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 23.3s

Alternatives: 16

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky))))

C

double code(double kx, double ky, double th) {
	return sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / Math.sin(ky));
}

Python

def code(kx, ky, th):
	return math.sin(th) / (math.hypot(math.sin(kx), math.sin(ky)) / math.sin(ky))

Julia

function code(kx, ky, th)
	return Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) / sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. *-commutative 94.6%

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. clear-num 94.6%

      \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   3. +-commutative 94.6%

      \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

   4. unpow2 94.6%

      \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

   5. unpow2 94.6%

      \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

   6. hypot-udef 99.6%

      \[\leadsto \sin th \cdot \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   7. un-div-inv 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

4. Final simplification 99.6%

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

Alternative 2: 47.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin ky) -0.25)
     t_1
     (if (<= (sin ky) -0.05)
       (/ (sin ky) (fabs (/ (sin ky) (sin th))))
       (if (<= (sin ky) -2e-8)
         t_1
         (if (<= (sin ky) 2e-124) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(ky) <= -0.25) {
		tmp = t_1;
	} else if (sin(ky) <= -0.05) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else if (sin(ky) <= -2e-8) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-124) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (sin(ky) <= (-0.25d0)) then
        tmp = t_1
    else if (sin(ky) <= (-0.05d0)) then
        tmp = sin(ky) / abs((sin(ky) / sin(th)))
    else if (sin(ky) <= (-2d-8)) then
        tmp = t_1
    else if (sin(ky) <= 2d-124) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(ky) <= -0.25) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -0.05) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else if (Math.sin(ky) <= -2e-8) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-124) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if math.sin(ky) <= -0.25:
		tmp = t_1
	elif math.sin(ky) <= -0.05:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	elif math.sin(ky) <= -2e-8:
		tmp = t_1
	elif math.sin(ky) <= 2e-124:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(ky) <= -0.25)
		tmp = t_1;
	elseif (sin(ky) <= -0.05)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	elseif (sin(ky) <= -2e-8)
		tmp = t_1;
	elseif (sin(ky) <= 2e-124)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (sin(ky) <= -0.25)
		tmp = t_1;
	elseif (sin(ky) <= -0.05)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	elseif (sin(ky) <= -2e-8)
		tmp = t_1;
	elseif (sin(ky) <= 2e-124)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.25], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.05], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-8], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-124], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.25 or -0.050000000000000003 < (sin.f64 ky) < -2e-8

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 2.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 2.6%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 2.6%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 2.6%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-sqr-sqrt 1.2%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      5. sqrt-unprod 21.4%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      6. pow2 21.4%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   6. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 21.4%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 34.8%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   8. Simplified 34.8%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.25 < (sin.f64 ky) < -0.050000000000000003

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 99.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 3.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th}} \cdot \sqrt{\frac{\sin ky}{\sin th}}}} \]

      2. sqrt-unprod 59.8%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      3. pow2 59.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]

   6. Applied egg-rr 59.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 59.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      2. rem-sqrt-square 74.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]

   8. Simplified 74.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]

   if -2e-8 < (sin.f64 ky) < 1.99999999999999987e-124

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. +-commutative 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-def 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

   4. Taylor expanded in ky around 0 58.8%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.99999999999999987e-124 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 58.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -5 \cdot 10^{-7} \lor \neg \left(\sin th \leq 10^{-17}\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= (sin th) -5e-7) (not (<= (sin th) 1e-17)))
     (/ (* (sin th) ky) t_1)
     (/ (sin ky) (/ t_1 th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((sin(th) <= -5e-7) || !(sin(th) <= 1e-17)) {
		tmp = (sin(th) * ky) / t_1;
	} else {
		tmp = sin(ky) / (t_1 / th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((Math.sin(th) <= -5e-7) || !(Math.sin(th) <= 1e-17)) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else {
		tmp = Math.sin(ky) / (t_1 / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (math.sin(th) <= -5e-7) or not (math.sin(th) <= 1e-17):
		tmp = (math.sin(th) * ky) / t_1
	else:
		tmp = math.sin(ky) / (t_1 / th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((sin(th) <= -5e-7) || !(sin(th) <= 1e-17))
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	else
		tmp = Float64(sin(ky) / Float64(t_1 / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((sin(th) <= -5e-7) || ~((sin(th) <= 1e-17)))
		tmp = (sin(th) * ky) / t_1;
	else
		tmp = sin(ky) / (t_1 / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[N[Sin[th], $MachinePrecision], -5e-7], N[Not[LessEqual[N[Sin[th], $MachinePrecision], 1e-17]], $MachinePrecision]], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 / th), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 th) < -4.99999999999999977e-7 or 1.00000000000000007e-17 < (sin.f64 th)

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. +-commutative 92.2%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      3. unpow2 92.2%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      4. unpow2 92.2%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Taylor expanded in ky around 0 54.8%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if -4.99999999999999977e-7 < (sin.f64 th) < 1.00000000000000007e-17

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 97.1%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 97.1%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in th around 0 99.7%

      \[\leadsto \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -5 \cdot 10^{-7} \lor \neg \left(\sin th \leq 10^{-17}\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \end{array} \]

Alternative 4: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (fabs (sin th))
   (if (<= (sin th) 2e-8)
     (* th (/ (sin ky) (hypot (sin ky) (sin kx))))
     (/ (sin ky) (fabs (/ (sin ky) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.05) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= 2e-8) {
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.05) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= 2e-8) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.05:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= 2e-8:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.05)
		tmp = abs(sin(th));
	elseif (sin(th) <= 2e-8)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.05)
		tmp = abs(sin(th));
	elseif (sin(th) <= 2e-8)
		tmp = th * (sin(ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 2e-8], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.050000000000000003

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 92.8%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 24.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 24.4%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 24.4%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 24.4%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      5. sqrt-unprod 26.5%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      6. pow2 26.5%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   6. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 26.5%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 26.5%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   8. Simplified 26.5%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.050000000000000003 < (sin.f64 th) < 2e-8

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. +-commutative 97.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 97.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 97.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-def 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

   4. Taylor expanded in th around 0 98.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]

   if 2e-8 < (sin.f64 th)

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 91.2%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 21.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 20.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th}} \cdot \sqrt{\frac{\sin ky}{\sin th}}}} \]

      2. sqrt-unprod 39.9%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      3. pow2 39.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]

   6. Applied egg-rr 39.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 39.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      2. rem-sqrt-square 44.3%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]

   8. Simplified 44.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \]

Alternative 5: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (fabs (sin th))
   (if (<= (sin th) 2e-8)
     (/ th (/ (hypot (sin kx) (sin ky)) (sin ky)))
     (/ (sin ky) (fabs (/ (sin ky) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.05) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= 2e-8) {
		tmp = th / (hypot(sin(kx), sin(ky)) / sin(ky));
	} else {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.05) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= 2e-8) {
		tmp = th / (Math.hypot(Math.sin(kx), Math.sin(ky)) / Math.sin(ky));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.05:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= 2e-8:
		tmp = th / (math.hypot(math.sin(kx), math.sin(ky)) / math.sin(ky))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.05)
		tmp = abs(sin(th));
	elseif (sin(th) <= 2e-8)
		tmp = Float64(th / Float64(hypot(sin(kx), sin(ky)) / sin(ky)));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.05)
		tmp = abs(sin(th));
	elseif (sin(th) <= 2e-8)
		tmp = th / (hypot(sin(kx), sin(ky)) / sin(ky));
	else
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 2e-8], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.050000000000000003

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 92.8%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 92.8%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 24.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 24.4%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 24.4%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 24.4%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      5. sqrt-unprod 26.5%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      6. pow2 26.5%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   6. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 26.5%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 26.5%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   8. Simplified 26.5%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.050000000000000003 < (sin.f64 th) < 2e-8

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 97.1%

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. +-commutative 97.1%

         \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      4. unpow2 97.1%

         \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      5. unpow2 97.1%

         \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      6. hypot-udef 99.6%

         \[\leadsto \sin th \cdot \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

      7. un-div-inv 99.7%

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   4. Taylor expanded in th around 0 98.9%

      \[\leadsto \frac{\color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   if 2e-8 < (sin.f64 th)

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 91.2%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 91.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 21.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 20.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th}} \cdot \sqrt{\frac{\sin ky}{\sin th}}}} \]

      2. sqrt-unprod 39.9%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      3. pow2 39.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]

   6. Applied egg-rr 39.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin th}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 39.9%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\sin ky}{\sin th} \cdot \frac{\sin ky}{\sin th}}}} \]

      2. rem-sqrt-square 44.3%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]

   8. Simplified 44.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\frac{\sin ky}{\sin th}\right|}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin th) (/ (sin ky) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	return sin(th) * (sin(ky) / hypot(sin(ky), sin(kx)));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
}

Python

def code(kx, ky, th):
	return math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), sin(kx))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) * (sin(ky) / hypot(sin(ky), sin(kx)));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. +-commutative 94.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   2. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   3. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   4. hypot-def 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

4. Final simplification 99.6%

   \[\leadsto \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

Alternative 7: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -0.07 \lor \neg \left(th \leq 0.08\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= th -0.07) (not (<= th 0.08)))
     (/ (* (sin th) ky) t_1)
     (/ (sin ky) (* t_1 (+ (* th 0.16666666666666666) (/ 1.0 th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((th <= -0.07) || !(th <= 0.08)) {
		tmp = (sin(th) * ky) / t_1;
	} else {
		tmp = sin(ky) / (t_1 * ((th * 0.16666666666666666) + (1.0 / th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((th <= -0.07) || !(th <= 0.08)) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else {
		tmp = Math.sin(ky) / (t_1 * ((th * 0.16666666666666666) + (1.0 / th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (th <= -0.07) or not (th <= 0.08):
		tmp = (math.sin(th) * ky) / t_1
	else:
		tmp = math.sin(ky) / (t_1 * ((th * 0.16666666666666666) + (1.0 / th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((th <= -0.07) || !(th <= 0.08))
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	else
		tmp = Float64(sin(ky) / Float64(t_1 * Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((th <= -0.07) || ~((th <= 0.08)))
		tmp = (sin(th) * ky) / t_1;
	else
		tmp = sin(ky) / (t_1 * ((th * 0.16666666666666666) + (1.0 / th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[th, -0.07], N[Not[LessEqual[th, 0.08]], $MachinePrecision]], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 * N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < -0.070000000000000007 or 0.0800000000000000017 < th

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 91.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. +-commutative 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      3. unpow2 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      4. unpow2 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Taylor expanded in ky around 0 53.5%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if -0.070000000000000007 < th < 0.0800000000000000017

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 97.2%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 97.2%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in th around 0 96.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. +-commutative 96.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. *-commutative 96.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \frac{1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 96.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. unpow2 96.7%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. hypot-def 99.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. associate-*r/ 99.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot 1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. associate-*l/ 99.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. associate-*r* 99.2%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -0.07 \lor \neg \left(th \leq 0.08\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \end{array} \]

Alternative 8: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq -0.22 \lor \neg \left(th \leq 0.23\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot 0.16666666666666666 + \frac{1}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= th -0.22) (not (<= th 0.23)))
   (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))
   (/
    (/ (sin ky) (hypot (sin kx) (sin ky)))
    (+ (* th 0.16666666666666666) (/ 1.0 th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((th <= -0.22) || !(th <= 0.23)) {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) / ((th * 0.16666666666666666) + (1.0 / th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((th <= -0.22) || !(th <= 0.23)) {
		tmp = (Math.sin(th) * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) / ((th * 0.16666666666666666) + (1.0 / th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (th <= -0.22) or not (th <= 0.23):
		tmp = (math.sin(th) * ky) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) / ((th * 0.16666666666666666) + (1.0 / th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((th <= -0.22) || !(th <= 0.23))
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) / Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((th <= -0.22) || ~((th <= 0.23)))
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	else
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) / ((th * 0.16666666666666666) + (1.0 / th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[th, -0.22], N[Not[LessEqual[th, 0.23]], $MachinePrecision]], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < -0.220000000000000001 or 0.23000000000000001 < th

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 91.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. +-commutative 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      3. unpow2 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      4. unpow2 91.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Taylor expanded in ky around 0 53.5%

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   if -0.220000000000000001 < th < 0.23000000000000001

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-*l/ 94.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. +-commutative 94.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      3. unpow2 94.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      4. unpow2 94.9%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      5. hypot-udef 95.7%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      6. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. div-inv 99.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin th}}} \]

      8. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{\sin th}}} \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{\sin th}}} \]

   4. Taylor expanded in th around 0 99.3%

      \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\color{blue}{0.16666666666666666 \cdot th + \frac{1}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -0.22 \lor \neg \left(th \leq 0.23\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot 0.16666666666666666 + \frac{1}{th}}\\ \end{array} \]

Alternative 9: 47.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-8)
   (fabs (sin th))
   (if (<= (sin ky) 2e-124) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-8) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-124) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-2d-8)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-124) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -2e-8) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-124) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-8:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-124:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-8)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-124)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-8)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-124)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-8], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-124], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2e-8

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 2.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 2.7%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 2.7%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 2.7%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-sqr-sqrt 1.3%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      5. sqrt-unprod 19.7%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      6. pow2 19.7%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   6. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 19.7%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 33.0%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   8. Simplified 33.0%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -2e-8 < (sin.f64 ky) < 1.99999999999999987e-124

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. +-commutative 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 87.1%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. hypot-def 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

   4. Taylor expanded in ky around 0 58.8%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.99999999999999987e-124 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 58.9%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 33.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \mathbf{elif}\;ky \leq 2.5 \cdot 10^{+139}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -2e-21)
     t_1
     (if (<= ky 1.12e-142)
       (/ ky (* (sin kx) (+ (* th 0.16666666666666666) (/ 1.0 th))))
       (if (<= ky 2.5e+139) (sin th) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -2e-21) {
		tmp = t_1;
	} else if (ky <= 1.12e-142) {
		tmp = ky / (sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	} else if (ky <= 2.5e+139) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (ky <= (-2d-21)) then
        tmp = t_1
    else if (ky <= 1.12d-142) then
        tmp = ky / (sin(kx) * ((th * 0.16666666666666666d0) + (1.0d0 / th)))
    else if (ky <= 2.5d+139) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -2e-21) {
		tmp = t_1;
	} else if (ky <= 1.12e-142) {
		tmp = ky / (Math.sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	} else if (ky <= 2.5e+139) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -2e-21:
		tmp = t_1
	elif ky <= 1.12e-142:
		tmp = ky / (math.sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)))
	elif ky <= 2.5e+139:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -2e-21)
		tmp = t_1;
	elseif (ky <= 1.12e-142)
		tmp = Float64(ky / Float64(sin(kx) * Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th))));
	elseif (ky <= 2.5e+139)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -2e-21)
		tmp = t_1;
	elseif (ky <= 1.12e-142)
		tmp = ky / (sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	elseif (ky <= 2.5e+139)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -2e-21], t$95$1, If[LessEqual[ky, 1.12e-142], N[(ky / N[(N[Sin[kx], $MachinePrecision] * N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.5e+139], N[Sin[th], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.99999999999999982e-21 or 2.50000000000000015e139 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 27.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 27.4%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 27.4%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 27.4%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-sqr-sqrt 14.0%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      5. sqrt-unprod 25.2%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      6. pow2 25.2%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   6. Applied egg-rr 25.2%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 25.2%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 36.3%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -1.99999999999999982e-21 < ky < 1.1199999999999999e-142

   1. Initial program 86.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 86.3%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 86.3%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in th around 0 49.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. +-commutative 49.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. *-commutative 49.4%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \frac{1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 49.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. unpow2 49.4%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. hypot-def 52.9%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. associate-*r/ 53.0%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot 1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-commutative 53.0%

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. associate-*l/ 52.9%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. associate-*r* 52.9%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   6. Simplified 53.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

   7. Taylor expanded in ky around 0 34.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx \cdot \left(0.16666666666666666 \cdot th + \frac{1}{th}\right)}} \]

   if 1.1199999999999999e-142 < ky < 2.50000000000000015e139

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 40.1%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \mathbf{elif}\;ky \leq 2.5 \cdot 10^{+139}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]

Alternative 11: 32.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -960000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 10^{-142}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -960000000.0)
   (sin th)
   (if (<= ky 1e-142)
     (/ ky (* (sin kx) (+ (* th 0.16666666666666666) (/ 1.0 th))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -960000000.0) {
		tmp = sin(th);
	} else if (ky <= 1e-142) {
		tmp = ky / (sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-960000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1d-142) then
        tmp = ky / (sin(kx) * ((th * 0.16666666666666666d0) + (1.0d0 / th)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -960000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1e-142) {
		tmp = ky / (Math.sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -960000000.0:
		tmp = math.sin(th)
	elif ky <= 1e-142:
		tmp = ky / (math.sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -960000000.0)
		tmp = sin(th);
	elseif (ky <= 1e-142)
		tmp = Float64(ky / Float64(sin(kx) * Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -960000000.0)
		tmp = sin(th);
	elseif (ky <= 1e-142)
		tmp = ky / (sin(kx) * ((th * 0.16666666666666666) + (1.0 / th)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -960000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1e-142], N[(ky / N[(N[Sin[kx], $MachinePrecision] * N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -9.6e8 or 1e-142 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 33.3%

      \[\leadsto \color{blue}{\sin th} \]

   if -9.6e8 < ky < 1e-142

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 87.0%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 87.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in th around 0 49.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. +-commutative 49.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. *-commutative 49.2%

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \frac{1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 49.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. unpow2 49.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. hypot-def 52.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      6. associate-*r/ 52.6%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot 1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      7. *-commutative 52.6%

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      8. associate-*l/ 52.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      9. associate-*r* 52.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   6. Simplified 52.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

   7. Taylor expanded in ky around 0 33.3%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx \cdot \left(0.16666666666666666 \cdot th + \frac{1}{th}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -960000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 10^{-142}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(th \cdot 0.16666666666666666 + \frac{1}{th}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 28.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5400000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7.6 \cdot 10^{-244}:\\ \;\;\;\;\log \left(1 + \left(th + 0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5400000.0)
   (sin th)
   (if (<= ky 7.6e-244) (log (+ 1.0 (+ th (* 0.5 (* th th))))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5400000.0) {
		tmp = sin(th);
	} else if (ky <= 7.6e-244) {
		tmp = log((1.0 + (th + (0.5 * (th * th)))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-5400000.0d0)) then
        tmp = sin(th)
    else if (ky <= 7.6d-244) then
        tmp = log((1.0d0 + (th + (0.5d0 * (th * th)))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5400000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 7.6e-244) {
		tmp = Math.log((1.0 + (th + (0.5 * (th * th)))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5400000.0:
		tmp = math.sin(th)
	elif ky <= 7.6e-244:
		tmp = math.log((1.0 + (th + (0.5 * (th * th)))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5400000.0)
		tmp = sin(th);
	elseif (ky <= 7.6e-244)
		tmp = log(Float64(1.0 + Float64(th + Float64(0.5 * Float64(th * th)))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5400000.0)
		tmp = sin(th);
	elseif (ky <= 7.6e-244)
		tmp = log((1.0 + (th + (0.5 * (th * th)))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5400000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 7.6e-244], N[Log[N[(1.0 + N[(th + N[(0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.4e6 or 7.6000000000000001e-244 < ky

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 32.7%

      \[\leadsto \color{blue}{\sin th} \]

   if -5.4e6 < ky < 7.6000000000000001e-244

   1. Initial program 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 89.0%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 89.0%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 3.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 3.7%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 3.7%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 3.7%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-log-exp 22.5%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   6. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Taylor expanded in th around 0 23.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(th + 0.5 \cdot {th}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. unpow2 23.2%

         \[\leadsto \log \left(1 + \left(th + 0.5 \cdot \color{blue}{\left(th \cdot th\right)}\right)\right) \]

   9. Simplified 23.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(th + 0.5 \cdot \left(th \cdot th\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5400000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7.6 \cdot 10^{-244}:\\ \;\;\;\;\log \left(1 + \left(th + 0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 28.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5400000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.1 \cdot 10^{-264}:\\ \;\;\;\;\log \left(th + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5400000.0)
   (sin th)
   (if (<= ky 3.1e-264) (log (+ th 1.0)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5400000.0) {
		tmp = sin(th);
	} else if (ky <= 3.1e-264) {
		tmp = log((th + 1.0));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-5400000.0d0)) then
        tmp = sin(th)
    else if (ky <= 3.1d-264) then
        tmp = log((th + 1.0d0))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5400000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 3.1e-264) {
		tmp = Math.log((th + 1.0));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5400000.0:
		tmp = math.sin(th)
	elif ky <= 3.1e-264:
		tmp = math.log((th + 1.0))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5400000.0)
		tmp = sin(th);
	elseif (ky <= 3.1e-264)
		tmp = log(Float64(th + 1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5400000.0)
		tmp = sin(th);
	elseif (ky <= 3.1e-264)
		tmp = log((th + 1.0));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5400000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.1e-264], N[Log[N[(th + 1.0), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.4e6 or 3.1000000000000002e-264 < ky

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0 31.9%

      \[\leadsto \color{blue}{\sin th} \]

   if -5.4e6 < ky < 3.1000000000000002e-264

   1. Initial program 89.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. associate-/r/ 89.6%

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

      2. +-commutative 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

      3. unpow2 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

      4. sqr-neg 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

      5. sin-neg 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

      6. sin-neg 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

      7. unpow2 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

      8. +-commutative 89.6%

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

   4. Taylor expanded in kx around 0 3.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 3.6%

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th} \]

      2. *-inverses 3.6%

         \[\leadsto \color{blue}{1} \cdot \sin th \]

      3. *-un-lft-identity 3.6%

         \[\leadsto \color{blue}{\sin th} \]

      4. add-log-exp 23.5%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   6. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Taylor expanded in th around 0 24.0%

      \[\leadsto \log \color{blue}{\left(1 + th\right)} \]
   8. Step-by-step derivation

      1. +-commutative 24.0%

         \[\leadsto \log \color{blue}{\left(th + 1\right)} \]

   9. Simplified 24.0%

      \[\leadsto \log \color{blue}{\left(th + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5400000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.1 \cdot 10^{-264}:\\ \;\;\;\;\log \left(th + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 23.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in kx around 0 23.3%

   \[\leadsto \color{blue}{\sin th} \]

3. Final simplification 23.3%

   \[\leadsto \sin th \]

Alternative 15: 14.6% accurate, 78.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{th \cdot 0.16666666666666666 + \frac{1}{th}} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ 1.0 (+ (* th 0.16666666666666666) (/ 1.0 th))))

C

double code(double kx, double ky, double th) {
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 / ((th * 0.16666666666666666d0) + (1.0d0 / th))
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
}

Python

def code(kx, ky, th):
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th))

Julia

function code(kx, ky, th)
	return Float64(1.0 / Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 / N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-/r/ 94.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

   2. +-commutative 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

   3. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

   4. sqr-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

   5. sin-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

   6. sin-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

   7. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

   8. +-commutative 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

4. Taylor expanded in th around 0 51.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
5. Step-by-step derivation

   1. +-commutative 51.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

   2. *-commutative 51.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \frac{1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   3. unpow2 51.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   4. unpow2 51.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   5. hypot-def 53.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \frac{1}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   6. associate-*r/ 53.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot 1}{th}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   7. *-commutative 53.1%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   8. associate-*l/ 53.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

   9. associate-*r* 53.0%

      \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

6. Simplified 53.0%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]

7. Taylor expanded in kx around 0 14.5%

   \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot th + \frac{1}{th}}} \]

8. Final simplification 14.5%

   \[\leadsto \frac{1}{th \cdot 0.16666666666666666 + \frac{1}{th}} \]

Alternative 16: 13.9% accurate, 709.0× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. associate-/r/ 94.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin th}}} \]

   2. +-commutative 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin th}} \]

   3. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin th}} \]

   4. sqr-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)}}}{\sin th}} \]

   5. sin-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right)}}{\sin th}} \]

   6. sin-neg 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)}}}{\sin th}} \]

   7. unpow2 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin \left(-kx\right)}^{2}}}}{\sin th}} \]

   8. +-commutative 94.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}}{\sin th}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

4. Taylor expanded in kx around 0 23.2%

   \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin ky}{\sin th}}} \]

5. Taylor expanded in th around 0 13.7%

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

6. Final simplification 13.7%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023272 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))