Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 20.6s

Alternatives: 15

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%

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

   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

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

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)))
    (+ (/ 1.0 th) (* th 0.16666666666666666)))))

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))) / ((1.0 / th) + (th * 0.16666666666666666));
	}
	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))) / ((1.0 / th) + (th * 0.16666666666666666));
	}
	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))) / ((1.0 / th) + (th * 0.16666666666666666))
	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(1.0 / th) + Float64(th * 0.16666666666666666)));
	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))) / ((1.0 / th) + (th * 0.16666666666666666));
	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[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $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%

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

   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%

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

      1. *-commutative 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 76.9%

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

Alternative 8: 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 9: 33.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -2 \cdot 10^{-163} \lor \neg \left(ky \leq 2.5 \cdot 10^{+139}\right):\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -2e-163) (not (<= ky 2.5e+139)))
   (fabs (sin th))
   (/ (sin th) (+ 1.0 (* 0.5 (* (/ kx ky) (/ kx ky)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -2e-163) || !(ky <= 2.5e+139)) {
		tmp = fabs(sin(th));
	} else {
		tmp = sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	}
	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 <= (-2d-163)) .or. (.not. (ky <= 2.5d+139))) then
        tmp = abs(sin(th))
    else
        tmp = sin(th) / (1.0d0 + (0.5d0 * ((kx / ky) * (kx / ky))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -2e-163) || !(ky <= 2.5e+139)) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = Math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -2e-163) or not (ky <= 2.5e+139):
		tmp = math.fabs(math.sin(th))
	else:
		tmp = math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -2e-163) || !(ky <= 2.5e+139))
		tmp = abs(sin(th));
	else
		tmp = Float64(sin(th) / Float64(1.0 + Float64(0.5 * Float64(Float64(kx / ky) * Float64(kx / ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -2e-163) || ~((ky <= 2.5e+139)))
		tmp = abs(sin(th));
	else
		tmp = sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -2e-163], N[Not[LessEqual[ky, 2.5e+139]], $MachinePrecision]], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(kx / ky), $MachinePrecision] * N[(kx / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.99999999999999985e-163 or 2.50000000000000015e139 < ky

   1. Initial program 99.2%

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

      1. associate-/r/ 99.1%

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

      2. +-commutative 99.1%

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

      3. unpow2 99.1%

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

      4. sqr-neg 99.1%

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

      5. sin-neg 99.1%

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

      6. sin-neg 99.1%

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

      7. unpow2 99.1%

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

      8. +-commutative 99.1%

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

   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 21.0%

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

      1. associate-/r/ 21.0%

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

      2. *-inverses 21.0%

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

      3. *-un-lft-identity 21.0%

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

      4. add-sqr-sqrt 10.7%

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

      5. sqrt-unprod 26.2%

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

      6. pow2 26.2%

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

   6. Applied egg-rr 26.2%

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

      1. unpow2 26.2%

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

      2. rem-sqrt-square 32.3%

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

   8. Simplified 32.3%

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

   if -1.99999999999999985e-163 < ky < 2.50000000000000015e139

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. clear-num 89.7%

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

      3. +-commutative 89.7%

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

      4. unpow2 89.7%

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

      5. unpow2 89.7%

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

      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. Taylor expanded in kx around 0 34.7%

      \[\leadsto \frac{\sin th}{\color{blue}{1 + 0.5 \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 34.7%

         \[\leadsto \frac{\sin th}{1 + \color{blue}{\frac{0.5 \cdot {kx}^{2}}{{\sin ky}^{2}}}} \]

      2. unpow2 34.7%

         \[\leadsto \frac{\sin th}{1 + \frac{0.5 \cdot \color{blue}{\left(kx \cdot kx\right)}}{{\sin ky}^{2}}} \]

   6. Simplified 34.7%

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}} \]

   7. Taylor expanded in ky around 0 34.9%

      \[\leadsto \frac{\sin th}{1 + \color{blue}{0.5 \cdot \frac{{kx}^{2}}{{ky}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 34.9%

         \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{\color{blue}{kx \cdot kx}}{{ky}^{2}}} \]

      2. unpow2 34.9%

         \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{kx \cdot kx}{\color{blue}{ky \cdot ky}}} \]

      3. times-frac 38.4%

         \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \color{blue}{\left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}} \]

   9. Simplified 38.4%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2 \cdot 10^{-163} \lor \neg \left(ky \leq 2.5 \cdot 10^{+139}\right):\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}\\ \end{array} \]

Alternative 10: 30.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ (sin th) (+ 1.0 (* 0.5 (* (/ kx ky) (/ kx ky))))))

C

double code(double kx, double ky, double th) {
	return sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
}

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) / (1.0d0 + (0.5d0 * ((kx / ky) * (kx / ky))))
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
}

Python

def code(kx, ky, th):
	return math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))))

Julia

function code(kx, ky, th)
	return Float64(sin(th) / Float64(1.0 + Float64(0.5 * Float64(Float64(kx / ky) * Float64(kx / ky)))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(kx / ky), $MachinePrecision] * N[(kx / ky), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in kx around 0 27.3%

   \[\leadsto \frac{\sin th}{\color{blue}{1 + 0.5 \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
5. Step-by-step derivation

   1. associate-*r/ 27.3%

      \[\leadsto \frac{\sin th}{1 + \color{blue}{\frac{0.5 \cdot {kx}^{2}}{{\sin ky}^{2}}}} \]

   2. unpow2 27.3%

      \[\leadsto \frac{\sin th}{1 + \frac{0.5 \cdot \color{blue}{\left(kx \cdot kx\right)}}{{\sin ky}^{2}}} \]

6. Simplified 27.3%

   \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}} \]

7. Taylor expanded in ky around 0 27.5%

   \[\leadsto \frac{\sin th}{1 + \color{blue}{0.5 \cdot \frac{{kx}^{2}}{{ky}^{2}}}} \]
8. Step-by-step derivation

   1. unpow2 27.5%

      \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{\color{blue}{kx \cdot kx}}{{ky}^{2}}} \]

   2. unpow2 27.5%

      \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{kx \cdot kx}{\color{blue}{ky \cdot ky}}} \]

   3. times-frac 29.8%

      \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \color{blue}{\left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}} \]

9. Simplified 29.8%

   \[\leadsto \frac{\sin th}{1 + \color{blue}{0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}} \]

10. Final simplification 29.8%

    \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)} \]

Alternative 11: 26.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.95e-42) (sin th) (+ (+ (sin th) 1.0) -1.0)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.95e-42) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) + 1.0) + -1.0;
	}
	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 (kx <= 1.95d-42) then
        tmp = sin(th)
    else
        tmp = (sin(th) + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.95e-42) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.95e-42:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) + 1.0) + -1.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.95e-42)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.95e-42)
		tmp = sin(th);
	else
		tmp = (sin(th) + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.95e-42], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.9500000000000001e-42

   1. Initial program 92.8%

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

   2. Taylor expanded in kx around 0 28.0%

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

   if 1.9500000000000001e-42 < kx

   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 10.4%

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

      1. associate-/r/ 10.4%

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

      2. *-inverses 10.4%

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

      3. *-un-lft-identity 10.4%

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

      4. expm1-log1p-u 10.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]

   6. Applied egg-rr 10.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-udef 19.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

      2. log1p-udef 19.4%

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

      3. add-exp-log 19.4%

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

      4. +-commutative 19.4%

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

   8. Applied egg-rr 19.4%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \end{array} \]

Alternative 12: 21.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.1e-32) (expm1 th) (if (<= ky 4.2e-140) (log1p 0.0) (expm1 th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.1e-32) {
		tmp = expm1(th);
	} else if (ky <= 4.2e-140) {
		tmp = log1p(0.0);
	} else {
		tmp = expm1(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.1e-32) {
		tmp = Math.expm1(th);
	} else if (ky <= 4.2e-140) {
		tmp = Math.log1p(0.0);
	} else {
		tmp = Math.expm1(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.1e-32:
		tmp = math.expm1(th)
	elif ky <= 4.2e-140:
		tmp = math.log1p(0.0)
	else:
		tmp = math.expm1(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.1e-32)
		tmp = expm1(th);
	elseif (ky <= 4.2e-140)
		tmp = log1p(0.0);
	else
		tmp = expm1(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.1e-32], N[(Exp[th] - 1), $MachinePrecision], If[LessEqual[ky, 4.2e-140], N[Log[1 + 0.0], $MachinePrecision], N[(Exp[th] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.1e-32 or 4.20000000000000035e-140 < 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 99.5%

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

   4. Taylor expanded in kx around 0 32.1%

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

      1. associate-/r/ 32.1%

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

      2. *-inverses 32.1%

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

      3. *-un-lft-identity 32.1%

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

      4. expm1-log1p-u 32.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]

   6. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]

   7. Taylor expanded in th around 0 19.8%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{th}\right) \]

   if -1.1e-32 < ky < 4.20000000000000035e-140

   1. Initial program 86.1%

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

      1. associate-/r/ 86.2%

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

      2. +-commutative 86.2%

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

      3. unpow2 86.2%

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

      4. sqr-neg 86.2%

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

      5. sin-neg 86.2%

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

      6. sin-neg 86.2%

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

      7. unpow2 86.2%

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

      8. +-commutative 86.2%

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

   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 8.0%

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

      1. associate-/r/ 8.0%

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

      2. *-inverses 8.0%

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

      3. *-un-lft-identity 8.0%

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

      4. log1p-expm1-u 8.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]

   6. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 7.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt[3]{\sin th} \cdot \sqrt[3]{\sin th}\right) \cdot \sqrt[3]{\sin th}}\right)\right) \]

      2. pow3 7.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\sin th}\right)}^{3}}\right)\right) \]

   8. Applied egg-rr 7.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\sin th}\right)}^{3}}\right)\right) \]

   9. Taylor expanded in th around 0 24.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \end{array} \]

Alternative 13: 29.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -960000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.25 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -960000000.0)
   (sin th)
   (if (<= ky 1.25e-242) (log1p 0.0) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -960000000.0) {
		tmp = sin(th);
	} else if (ky <= 1.25e-242) {
		tmp = log1p(0.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -960000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.25e-242) {
		tmp = Math.log1p(0.0);
	} 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 <= 1.25e-242:
		tmp = math.log1p(0.0)
	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 <= 1.25e-242)
		tmp = log1p(0.0);
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -960000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.25e-242], N[Log[1 + 0.0], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -9.6e8 or 1.25e-242 < 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 32.8%

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

   if -9.6e8 < ky < 1.25e-242

   1. Initial program 89.2%

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

      1. associate-/r/ 89.1%

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

      2. +-commutative 89.1%

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

      3. unpow2 89.1%

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

      4. sqr-neg 89.1%

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

      5. sin-neg 89.1%

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

      6. sin-neg 89.1%

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

      7. unpow2 89.1%

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

      8. +-commutative 89.1%

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

   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. log1p-expm1-u 3.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]

   6. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 3.7%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt[3]{\sin th} \cdot \sqrt[3]{\sin th}\right) \cdot \sqrt[3]{\sin th}}\right)\right) \]

      2. pow3 3.7%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\sin th}\right)}^{3}}\right)\right) \]

   8. Applied egg-rr 3.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(\sqrt[3]{\sin th}\right)}^{3}}\right)\right) \]

   9. Taylor expanded in th around 0 23.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -960000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.25 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 14.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(th\right) \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (expm1 th))

C

double code(double kx, double ky, double th) {
	return expm1(th);
}

Java

public static double code(double kx, double ky, double th) {
	return Math.expm1(th);
}

Python

def code(kx, ky, th):
	return math.expm1(th)

Julia

function code(kx, ky, th)
	return expm1(th)
end

Wolfram

code[kx_, ky_, th_] := N[(Exp[th] - 1), $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 kx around 0 23.2%

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

   1. associate-/r/ 23.3%

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

   2. *-inverses 23.3%

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

   3. *-un-lft-identity 23.3%

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

   4. expm1-log1p-u 23.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]

6. Applied egg-rr 23.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]

7. Taylor expanded in th around 0 14.3%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{th}\right) \]

8. Final simplification 14.3%

   \[\leadsto \mathsf{expm1}\left(th\right) \]

Alternative 15: 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)))