Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 28.0s

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: 94.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. +-commutative 94.2%

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

   2. unpow2 94.2%

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

   3. unpow2 94.2%

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

   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. Final simplification 99.7%

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

Alternative 2: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \sin ky \cdot \frac{th}{t_1}\\ \mathbf{if}\;\sin ky \leq -0.018:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.8425:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (* (sin ky) (/ th t_1))))
   (if (<= (sin ky) -0.018)
     t_2
     (if (<= (sin ky) 1e-9)
       (* (sin th) (/ ky t_1))
       (if (<= (sin ky) 0.8425) (sin th) t_2)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = sin(ky) * (th / t_1);
	double tmp;
	if (sin(ky) <= -0.018) {
		tmp = t_2;
	} else if (sin(ky) <= 1e-9) {
		tmp = sin(th) * (ky / t_1);
	} else if (sin(ky) <= 0.8425) {
		tmp = sin(th);
	} else {
		tmp = t_2;
	}
	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 t_2 = Math.sin(ky) * (th / t_1);
	double tmp;
	if (Math.sin(ky) <= -0.018) {
		tmp = t_2;
	} else if (Math.sin(ky) <= 1e-9) {
		tmp = Math.sin(th) * (ky / t_1);
	} else if (Math.sin(ky) <= 0.8425) {
		tmp = Math.sin(th);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.sin(ky) * (th / t_1)
	tmp = 0
	if math.sin(ky) <= -0.018:
		tmp = t_2
	elif math.sin(ky) <= 1e-9:
		tmp = math.sin(th) * (ky / t_1)
	elif math.sin(ky) <= 0.8425:
		tmp = math.sin(th)
	else:
		tmp = t_2
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(sin(ky) * Float64(th / t_1))
	tmp = 0.0
	if (sin(ky) <= -0.018)
		tmp = t_2;
	elseif (sin(ky) <= 1e-9)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	elseif (sin(ky) <= 0.8425)
		tmp = sin(th);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = sin(ky) * (th / t_1);
	tmp = 0.0;
	if (sin(ky) <= -0.018)
		tmp = t_2;
	elseif (sin(ky) <= 1e-9)
		tmp = sin(th) * (ky / t_1);
	elseif (sin(ky) <= 0.8425)
		tmp = sin(th);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.018], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.8425], N[Sin[th], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0179999999999999986 or 0.84250000000000003 < (sin.f64 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 th around 0 62.3%

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

      1. associate-*r/ 62.4%

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

      2. +-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. hypot-def 62.4%

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

      6. *-rgt-identity 62.4%

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

      7. hypot-def 62.4%

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

      8. unpow2 62.4%

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

      9. unpow2 62.4%

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

      10. +-commutative 62.4%

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

      11. unpow2 62.4%

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

      12. unpow2 62.4%

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

      13. hypot-def 62.4%

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

   6. Simplified 62.4%

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

      1. clear-num 62.4%

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

      2. associate-/r/ 62.2%

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

      3. clear-num 62.3%

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

   8. Applied egg-rr 62.3%

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

   if -0.0179999999999999986 < (sin.f64 ky) < 1.00000000000000006e-9

   1. Initial program 89.4%

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

      1. associate-*l/ 85.7%

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

      2. +-commutative 85.7%

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

      3. unpow2 85.7%

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

      4. unpow2 85.7%

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

      5. hypot-def 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in ky around 0 92.4%

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

      1. expm1-log1p-u 92.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 38.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. associate-/l* 38.8%

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

      4. hypot-udef 32.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{ky}}\right)} - 1 \]

      5. +-commutative 32.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{ky}}\right)} - 1 \]

      6. hypot-udef 38.8%

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

   6. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)\right)} \]

      2. expm1-log1p 98.7%

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

      3. associate-/r/ 98.7%

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

      4. associate-*l/ 92.4%

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

      5. associate-*r/ 98.8%

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

      6. hypot-def 88.5%

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

      7. unpow2 88.5%

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

      8. unpow2 88.5%

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

      9. +-commutative 88.5%

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

      10. unpow2 88.5%

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

      11. unpow2 88.5%

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

      12. hypot-def 98.8%

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

   8. Simplified 98.8%

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

   if 1.00000000000000006e-9 < (sin.f64 ky) < 0.84250000000000003

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 66.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.018:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.8425:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1} \cdot th\\ \mathbf{if}\;\sin ky \leq -0.018:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.8425:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (* (/ (sin ky) t_1) th)))
   (if (<= (sin ky) -0.018)
     t_2
     (if (<= (sin ky) 1e-9)
       (* (sin th) (/ ky t_1))
       (if (<= (sin ky) 0.8425) (sin th) t_2)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) / t_1) * th;
	double tmp;
	if (sin(ky) <= -0.018) {
		tmp = t_2;
	} else if (sin(ky) <= 1e-9) {
		tmp = sin(th) * (ky / t_1);
	} else if (sin(ky) <= 0.8425) {
		tmp = sin(th);
	} else {
		tmp = t_2;
	}
	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 t_2 = (Math.sin(ky) / t_1) * th;
	double tmp;
	if (Math.sin(ky) <= -0.018) {
		tmp = t_2;
	} else if (Math.sin(ky) <= 1e-9) {
		tmp = Math.sin(th) * (ky / t_1);
	} else if (Math.sin(ky) <= 0.8425) {
		tmp = Math.sin(th);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	t_2 = (math.sin(ky) / t_1) * th
	tmp = 0
	if math.sin(ky) <= -0.018:
		tmp = t_2
	elif math.sin(ky) <= 1e-9:
		tmp = math.sin(th) * (ky / t_1)
	elif math.sin(ky) <= 0.8425:
		tmp = math.sin(th)
	else:
		tmp = t_2
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) / t_1) * th)
	tmp = 0.0
	if (sin(ky) <= -0.018)
		tmp = t_2;
	elseif (sin(ky) <= 1e-9)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	elseif (sin(ky) <= 0.8425)
		tmp = sin(th);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = (sin(ky) / t_1) * th;
	tmp = 0.0;
	if (sin(ky) <= -0.018)
		tmp = t_2;
	elseif (sin(ky) <= 1e-9)
		tmp = sin(th) * (ky / t_1);
	elseif (sin(ky) <= 0.8425)
		tmp = sin(th);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.018], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.8425], N[Sin[th], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0179999999999999986 or 0.84250000000000003 < (sin.f64 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 th around 0 62.3%

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

      1. associate-*r/ 62.4%

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

      2. +-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. hypot-def 62.4%

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

      6. *-rgt-identity 62.4%

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

      7. hypot-def 62.4%

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

      8. unpow2 62.4%

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

      9. unpow2 62.4%

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

      10. +-commutative 62.4%

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

      11. unpow2 62.4%

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

      12. unpow2 62.4%

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

      13. hypot-def 62.4%

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

   6. Simplified 62.4%

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

      1. associate-/r/ 62.5%

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

   8. Applied egg-rr 62.5%

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

   if -0.0179999999999999986 < (sin.f64 ky) < 1.00000000000000006e-9

   1. Initial program 89.4%

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

      1. associate-*l/ 85.7%

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

      2. +-commutative 85.7%

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

      3. unpow2 85.7%

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

      4. unpow2 85.7%

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

      5. hypot-def 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in ky around 0 92.4%

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

      1. expm1-log1p-u 92.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 38.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. associate-/l* 38.8%

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

      4. hypot-udef 32.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{ky}}\right)} - 1 \]

      5. +-commutative 32.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{ky}}\right)} - 1 \]

      6. hypot-udef 38.8%

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

   6. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)\right)} \]

      2. expm1-log1p 98.7%

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

      3. associate-/r/ 98.7%

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

      4. associate-*l/ 92.4%

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

      5. associate-*r/ 98.8%

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

      6. hypot-def 88.5%

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

      7. unpow2 88.5%

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

      8. unpow2 88.5%

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

      9. +-commutative 88.5%

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

      10. unpow2 88.5%

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

      11. unpow2 88.5%

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

      12. hypot-def 98.8%

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

   8. Simplified 98.8%

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

   if 1.00000000000000006e-9 < (sin.f64 ky) < 0.84250000000000003

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 66.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.018:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.8425:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \]

Alternative 4: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -5 \cdot 10^{-5} \lor \neg \left(\sin th \leq 0.005\right):\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= (sin th) -5e-5) (not (<= (sin th) 0.005)))
   (/
    (sin th)
    (* (hypot (sin kx) (sin ky)) (+ (* ky 0.16666666666666666) (/ 1.0 ky))))
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) <= -5e-5) || !(sin(th) <= 0.005)) {
		tmp = sin(th) / (hypot(sin(kx), sin(ky)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(th) <= -5e-5) || !(Math.sin(th) <= 0.005)) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(th) <= -5e-5) or not (math.sin(th) <= 0.005):
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.sin(ky)) * ((ky * 0.16666666666666666) + (1.0 / ky)))
	else:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(th) <= -5e-5) || !(sin(th) <= 0.005))
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(th) <= -5e-5) || ~((sin(th) <= 0.005)))
		tmp = sin(th) / (hypot(sin(kx), sin(ky)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	else
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[N[Sin[th], $MachinePrecision], -5e-5], N[Not[LessEqual[N[Sin[th], $MachinePrecision], 0.005]], $MachinePrecision]], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 th) < -5.00000000000000024e-5 or 0.0050000000000000001 < (sin.f64 th)

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. unpow2 92.3%

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

      3. unpow2 92.3%

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

      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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.5%

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

      4. hypot-udef 92.2%

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

      5. unpow2 92.2%

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

      6. unpow2 92.2%

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

      7. +-commutative 92.2%

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

      8. unpow2 92.2%

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

      9. unpow2 92.2%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in ky around 0 60.7%

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

   if -5.00000000000000024e-5 < (sin.f64 th) < 0.0050000000000000001

   1. Initial program 95.8%

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

      1. associate-/r/ 95.7%

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

      2. +-commutative 95.7%

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

      3. unpow2 95.7%

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

      4. sqr-neg 95.7%

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

      5. sin-neg 95.7%

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

      6. sin-neg 95.7%

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

      7. unpow2 95.7%

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

      8. +-commutative 95.7%

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

   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 94.7%

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

      1. associate-*r/ 94.8%

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

      2. +-commutative 94.8%

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

      3. unpow2 94.8%

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

      4. unpow2 94.8%

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

      5. hypot-def 98.7%

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

      6. *-rgt-identity 98.7%

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

      7. hypot-def 94.8%

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

      8. unpow2 94.8%

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

      9. unpow2 94.8%

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

      10. +-commutative 94.8%

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

      11. unpow2 94.8%

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

      12. unpow2 94.8%

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

      13. hypot-def 98.7%

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

   6. Simplified 98.7%

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

      1. associate-/r/ 98.9%

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

   8. Applied egg-rr 98.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -5 \cdot 10^{-5} \lor \neg \left(\sin th \leq 0.005\right):\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \]

Alternative 5: 44.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) 4e-58) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 4e-58) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	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(kx) <= (-0.04d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= 4d-58) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 4e-58) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 4e-58:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 4e-58)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 4e-58)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-58], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 15.0%

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

      1. add-sqr-sqrt 12.3%

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

      2. sqrt-unprod 22.9%

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

      3. pow2 22.9%

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

      4. div-inv 22.8%

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

      5. associate-*l* 22.8%

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

      6. associate-/r/ 22.8%

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

      7. *-un-lft-identity 22.8%

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

      8. associate-*l/ 22.8%

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

      9. associate-/r* 22.9%

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

      10. clear-num 22.9%

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

      11. /-rgt-identity 22.9%

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

   4. Applied egg-rr 22.9%

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

      1. associate-*r/ 22.9%

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

      2. *-commutative 22.9%

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

      3. unpow2 22.9%

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

      4. rem-sqrt-square 38.8%

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

      5. *-commutative 38.8%

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

      6. associate-*r/ 38.9%

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

   6. Simplified 38.9%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 4.0000000000000001e-58

   1. Initial program 87.9%

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

      1. +-commutative 87.9%

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

      2. unpow2 87.9%

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

      3. unpow2 87.9%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 46.9%

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

   if 4.0000000000000001e-58 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 60.0%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 6: 50.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{-\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* (sin ky) (/ (- (sin th)) (sin kx)))
   (if (<= (sin kx) 4e-58) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = sin(ky) * (-sin(th) / sin(kx));
	} else if (sin(kx) <= 4e-58) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	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(kx) <= (-0.02d0)) then
        tmp = sin(ky) * (-sin(th) / sin(kx))
    else if (sin(kx) <= 4d-58) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.sin(ky) * (-Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(kx) <= 4e-58) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.sin(ky) * (-math.sin(th) / math.sin(kx))
	elif math.sin(kx) <= 4e-58:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(sin(ky) * Float64(Float64(-sin(th)) / sin(kx)));
	elseif (sin(kx) <= 4e-58)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = sin(ky) * (-sin(th) / sin(kx));
	elseif (sin(kx) <= 4e-58)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(N[Sin[ky], $MachinePrecision] * N[((-N[Sin[th], $MachinePrecision]) / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-58], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. associate-/r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. sqr-neg 99.4%

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

      5. sin-neg 99.4%

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

      6. sin-neg 99.4%

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

      7. unpow2 99.4%

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

      8. +-commutative 99.4%

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

   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 ky around 0 14.9%

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

      1. frac-2neg 14.9%

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

      2. neg-sub0 14.9%

         \[\leadsto \frac{\color{blue}{0 - \sin ky}}{-\frac{\sin kx}{\sin th}} \]

      3. metadata-eval 14.9%

         \[\leadsto \frac{\color{blue}{\log 1} - \sin ky}{-\frac{\sin kx}{\sin th}} \]

      4. div-sub 14.9%

         \[\leadsto \color{blue}{\frac{\log 1}{-\frac{\sin kx}{\sin th}} - \frac{\sin ky}{-\frac{\sin kx}{\sin th}}} \]

      5. metadata-eval 14.9%

         \[\leadsto \frac{\color{blue}{0}}{-\frac{\sin kx}{\sin th}} - \frac{\sin ky}{-\frac{\sin kx}{\sin th}} \]

      6. distribute-neg-frac 14.9%

         \[\leadsto \frac{0}{\color{blue}{\frac{-\sin kx}{\sin th}}} - \frac{\sin ky}{-\frac{\sin kx}{\sin th}} \]

      7. add-sqr-sqrt 5.7%

         \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \frac{\color{blue}{\sqrt{\sin ky} \cdot \sqrt{\sin ky}}}{-\frac{\sin kx}{\sin th}} \]

      8. sqrt-unprod 27.3%

         \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky}}}{-\frac{\sin kx}{\sin th}} \]

      9. sqr-neg 27.3%

         \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \frac{\sqrt{\color{blue}{\left(-\sin ky\right) \cdot \left(-\sin ky\right)}}}{-\frac{\sin kx}{\sin th}} \]

      10. sqrt-unprod 29.6%

          \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \frac{\color{blue}{\sqrt{-\sin ky} \cdot \sqrt{-\sin ky}}}{-\frac{\sin kx}{\sin th}} \]

      11. add-sqr-sqrt 62.0%

          \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \frac{\color{blue}{-\sin ky}}{-\frac{\sin kx}{\sin th}} \]

      12. frac-2neg 62.0%

          \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \color{blue}{\frac{\sin ky}{\frac{\sin kx}{\sin th}}} \]

      13. associate-/r/ 62.0%

          \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \color{blue}{\frac{\sin ky}{\sin kx} \cdot \sin th} \]

      14. *-commutative 62.0%

          \[\leadsto \frac{0}{\frac{-\sin kx}{\sin th}} - \color{blue}{\sin th \cdot \frac{\sin ky}{\sin kx}} \]

   6. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{0}{\frac{-\sin kx}{\sin th}} - \sin th \cdot \frac{\sin ky}{\sin kx}} \]
   7. Step-by-step derivation

      1. div0 62.0%

         \[\leadsto \color{blue}{0} - \sin th \cdot \frac{\sin ky}{\sin kx} \]

      2. neg-sub0 62.0%

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

      3. associate-*r/ 62.1%

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

      4. associate-*l/ 62.1%

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

      5. *-commutative 62.1%

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

      6. distribute-rgt-neg-in 62.1%

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

      7. distribute-frac-neg 62.1%

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

   8. Simplified 62.1%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 4.0000000000000001e-58

   1. Initial program 87.9%

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

      1. +-commutative 87.9%

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

      2. unpow2 87.9%

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

      3. unpow2 87.9%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 46.9%

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

   if 4.0000000000000001e-58 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 60.0%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{-\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 7: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-9)
   (* (sin th) (/ ky (hypot (sin ky) (sin kx))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-9) {
		tmp = sin(th) * (ky / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-9) {
		tmp = Math.sin(th) * (ky / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-9:
		tmp = math.sin(th) * (ky / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-9)
		tmp = Float64(sin(th) * Float64(ky / hypot(sin(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) <= 1e-9)
		tmp = sin(th) * (ky / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 1.00000000000000006e-9

   1. Initial program 92.4%

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

      1. associate-*l/ 89.8%

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

      2. +-commutative 89.8%

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

      3. unpow2 89.8%

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

      4. unpow2 89.8%

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

      5. hypot-def 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in ky around 0 66.0%

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

      1. expm1-log1p-u 65.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 28.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. associate-/l* 28.2%

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

      4. hypot-udef 23.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{ky}}\right)} - 1 \]

      5. +-commutative 23.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{ky}}\right)} - 1 \]

      6. hypot-udef 28.2%

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

   6. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 70.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}\right)\right)} \]

      2. expm1-log1p 70.4%

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

      3. associate-/r/ 70.4%

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

      4. associate-*l/ 66.0%

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

      5. associate-*r/ 70.5%

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

      6. hypot-def 63.2%

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

      7. unpow2 63.2%

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

      8. unpow2 63.2%

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

      9. +-commutative 63.2%

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

      10. unpow2 63.2%

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

      11. unpow2 63.2%

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

      12. hypot-def 70.5%

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

   8. Simplified 70.5%

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

   if 1.00000000000000006e-9 < (sin.f64 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 59.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 41.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-43}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-43) (sin th) (/ (sin th) (/ (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 1e-43) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(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 (sin(kx) <= (-0.04d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= 1d-43) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 1e-43) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 1e-43:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-43)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-43)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-43], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 15.0%

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

      1. add-sqr-sqrt 12.3%

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

      2. sqrt-unprod 22.9%

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

      3. pow2 22.9%

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

      4. div-inv 22.8%

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

      5. associate-*l* 22.8%

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

      6. associate-/r/ 22.8%

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

      7. *-un-lft-identity 22.8%

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

      8. associate-*l/ 22.8%

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

      9. associate-/r* 22.9%

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

      10. clear-num 22.9%

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

      11. /-rgt-identity 22.9%

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

   4. Applied egg-rr 22.9%

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

      1. associate-*r/ 22.9%

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

      2. *-commutative 22.9%

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

      3. unpow2 22.9%

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

      4. rem-sqrt-square 38.8%

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

      5. *-commutative 38.8%

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

      6. associate-*r/ 38.9%

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

   6. Simplified 38.9%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 1.00000000000000008e-43

   1. Initial program 88.2%

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

      1. +-commutative 88.2%

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

      2. unpow2 88.2%

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

      3. unpow2 88.2%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 46.7%

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

   if 1.00000000000000008e-43 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

      4. hypot-udef 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. +-commutative 99.3%

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

      8. unpow2 99.3%

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

      9. unpow2 99.3%

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

      10. hypot-def 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in ky around 0 50.1%

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

      1. associate-/l* 51.5%

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

   8. Simplified 51.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-43}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 9: 41.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-94) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-94) {
		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) <= 4d-94) 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) <= 4e-94) {
		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) <= 4e-94:
		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) <= 4e-94)
		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) <= 4e-94)
		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], 4e-94], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.9999999999999998e-94

   1. Initial program 91.4%

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

   2. Taylor expanded in ky around 0 29.3%

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

   if 3.9999999999999998e-94 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 60.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 41.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.36 \cdot 10^{-84}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -3.2)
   (sin th)
   (if (<= ky 1.36e-84) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -3.2) {
		tmp = sin(th);
	} else if (ky <= 1.36e-84) {
		tmp = ky * (sin(th) / 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 (ky <= (-3.2d0)) then
        tmp = sin(th)
    else if (ky <= 1.36d-84) then
        tmp = ky * (sin(th) / 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 (ky <= -3.2) {
		tmp = Math.sin(th);
	} else if (ky <= 1.36e-84) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -3.2:
		tmp = math.sin(th)
	elif ky <= 1.36e-84:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -3.2)
		tmp = sin(th);
	elseif (ky <= 1.36e-84)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -3.2)
		tmp = sin(th);
	elseif (ky <= 1.36e-84)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -3.2], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.36e-84], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.2000000000000002 or 1.36e-84 < 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 37.0%

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

   if -3.2000000000000002 < ky < 1.36e-84

   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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. hypot-udef 87.0%

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

      5. unpow2 87.0%

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

      6. unpow2 87.0%

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

      7. +-commutative 87.0%

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

      8. unpow2 87.0%

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

      9. unpow2 87.0%

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

      10. hypot-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r/ 43.0%

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

   10. Simplified 43.0%

       \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{\sin kx}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.36 \cdot 10^{-84}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 33.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-107}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -6200000000.0)
   (sin th)
   (if (<= ky 7e-107) (* (sin th) (/ ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6200000000.0) {
		tmp = sin(th);
	} else if (ky <= 7e-107) {
		tmp = sin(th) * (ky / 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 (ky <= (-6200000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 7d-107) then
        tmp = sin(th) * (ky / 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 (ky <= -6200000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 7e-107) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -6200000000.0:
		tmp = math.sin(th)
	elif ky <= 7e-107:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 7e-107)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 7e-107)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -6200000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 7e-107], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.2e9 or 6.99999999999999971e-107 < 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 37.5%

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

   if -6.2e9 < ky < 6.99999999999999971e-107

   1. Initial program 86.7%

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

   2. Taylor expanded in ky around 0 42.6%

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

   3. Taylor expanded in kx around 0 30.6%

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7 \cdot 10^{-107}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 27.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.15 \cdot 10^{-43}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.15e-43) (sin th) (/ ky (/ (sin kx) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.15e-43) {
		tmp = sin(th);
	} else {
		tmp = ky / (sin(kx) / 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 (kx <= 1.15d-43) then
        tmp = sin(th)
    else
        tmp = ky / (sin(kx) / th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.15e-43) {
		tmp = Math.sin(th);
	} else {
		tmp = ky / (Math.sin(kx) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.15e-43:
		tmp = math.sin(th)
	else:
		tmp = ky / (math.sin(kx) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.15e-43)
		tmp = sin(th);
	else
		tmp = Float64(ky / Float64(sin(kx) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.15e-43)
		tmp = sin(th);
	else
		tmp = ky / (sin(kx) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.15e-43], N[Sin[th], $MachinePrecision], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.1499999999999999e-43

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. unpow2 91.9%

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

      3. unpow2 91.9%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 34.0%

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

   if 1.1499999999999999e-43 < kx

   1. Initial program 99.4%

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

      1. associate-*l/ 98.3%

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

      2. +-commutative 98.3%

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

      3. unpow2 98.3%

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

      4. unpow2 98.3%

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

      5. hypot-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in ky around 0 48.2%

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

   5. Taylor expanded in th around 0 28.6%

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

   6. Taylor expanded in ky around 0 23.5%

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

      1. associate-/l* 24.7%

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

   8. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.15 \cdot 10^{-43}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \]

Alternative 13: 31.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -6200000000.0)
   (sin th)
   (if (<= ky 3.5e-214) (/ ky (/ kx th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6200000000.0) {
		tmp = sin(th);
	} else if (ky <= 3.5e-214) {
		tmp = ky / (kx / 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 <= (-6200000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 3.5d-214) then
        tmp = ky / (kx / 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 <= -6200000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 3.5e-214) {
		tmp = ky / (kx / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -6200000000.0:
		tmp = math.sin(th)
	elif ky <= 3.5e-214:
		tmp = ky / (kx / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 3.5e-214)
		tmp = Float64(ky / Float64(kx / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 3.5e-214)
		tmp = ky / (kx / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -6200000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.5e-214], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.2e9 or 3.5e-214 < ky

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. unpow2 97.7%

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

      3. unpow2 97.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 36.0%

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

   if -6.2e9 < ky < 3.5e-214

   1. Initial program 87.0%

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

   2. Taylor expanded in ky around 0 43.9%

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

   3. Taylor expanded in kx around 0 32.4%

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

   4. Taylor expanded in th around 0 22.7%

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

      1. *-commutative 22.7%

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

      2. associate-/l* 27.4%

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

   6. Simplified 27.4%

      \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 21.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.2 \cdot 10^{-69}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.2e-69) th (if (<= ky 8.5e-101) (/ ky (/ kx th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.2e-69) {
		tmp = th;
	} else if (ky <= 8.5e-101) {
		tmp = ky / (kx / th);
	} else {
		tmp = 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 <= (-5.2d-69)) then
        tmp = th
    else if (ky <= 8.5d-101) then
        tmp = ky / (kx / th)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.2e-69) {
		tmp = th;
	} else if (ky <= 8.5e-101) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.2e-69:
		tmp = th
	elif ky <= 8.5e-101:
		tmp = ky / (kx / th)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.2e-69)
		tmp = th;
	elseif (ky <= 8.5e-101)
		tmp = Float64(ky / Float64(kx / th));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.2e-69)
		tmp = th;
	elseif (ky <= 8.5e-101)
		tmp = ky / (kx / th);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.2e-69], th, If[LessEqual[ky, 8.5e-101], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.2000000000000004e-69 or 8.49999999999999941e-101 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 th around 0 57.6%

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

      1. associate-*r/ 57.8%

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

      2. +-commutative 57.8%

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

      3. unpow2 57.8%

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

      4. unpow2 57.8%

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

      5. hypot-def 57.8%

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

      6. *-rgt-identity 57.8%

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

      7. hypot-def 57.8%

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

      8. unpow2 57.8%

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

      9. unpow2 57.8%

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

      10. +-commutative 57.8%

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

      11. unpow2 57.8%

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

      12. unpow2 57.8%

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

      13. hypot-def 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in kx around 0 18.2%

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

   if -5.2000000000000004e-69 < ky < 8.49999999999999941e-101

   1. Initial program 85.2%

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

   2. Taylor expanded in ky around 0 47.2%

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

   3. Taylor expanded in kx around 0 33.7%

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

   4. Taylor expanded in th around 0 25.5%

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

      1. *-commutative 25.5%

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

      2. associate-/l* 29.7%

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

   6. Simplified 29.7%

      \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.2 \cdot 10^{-69}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 15: 14.0% 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 94.2%

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

   1. associate-/r/ 94.1%

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

   2. +-commutative 94.1%

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

   3. unpow2 94.1%

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

   4. sqr-neg 94.1%

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

   5. sin-neg 94.1%

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

   6. sin-neg 94.1%

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

   7. unpow2 94.1%

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

   8. +-commutative 94.1%

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

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 th around 0 52.1%

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

   1. associate-*r/ 52.1%

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

   2. +-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. hypot-def 54.5%

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

   6. *-rgt-identity 54.5%

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

   7. hypot-def 52.1%

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

   8. unpow2 52.1%

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

   9. unpow2 52.1%

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

   10. +-commutative 52.1%

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

   11. unpow2 52.1%

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

   12. unpow2 52.1%

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

   13. hypot-def 54.5%

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

6. Simplified 54.5%

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

7. Taylor expanded in kx around 0 13.8%

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

8. Final simplification 13.8%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023275 
(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)))