Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 31.5s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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 97.0%

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

   1. +-commutative 97.0%

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

   2. unpow2 97.0%

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

   3. unpow2 97.0%

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

   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: 49.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\frac{-1}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ ky (sin kx))))
   (if (<= (sin ky) -4e-28)
     (/ 1.0 (fabs (fma th 0.16666666666666666 (/ 1.0 th))))
     (if (<= (sin ky) -2e-301)
       (* t_1 (/ -1.0 (/ -1.0 (sin th))))
       (if (<= (sin ky) 5e-26) (* (sin th) (fabs t_1)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky / sin(kx);
	double tmp;
	if (sin(ky) <= -4e-28) {
		tmp = 1.0 / fabs(fma(th, 0.16666666666666666, (1.0 / th)));
	} else if (sin(ky) <= -2e-301) {
		tmp = t_1 * (-1.0 / (-1.0 / sin(th)));
	} else if (sin(ky) <= 5e-26) {
		tmp = sin(th) * fabs(t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky / sin(kx))
	tmp = 0.0
	if (sin(ky) <= -4e-28)
		tmp = Float64(1.0 / abs(fma(th, 0.16666666666666666, Float64(1.0 / th))));
	elseif (sin(ky) <= -2e-301)
		tmp = Float64(t_1 * Float64(-1.0 / Float64(-1.0 / sin(th))));
	elseif (sin(ky) <= 5e-26)
		tmp = Float64(sin(th) * abs(t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-28], N[(1.0 / N[Abs[N[(th * 0.16666666666666666 + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-301], N[(t$95$1 * N[(-1.0 / N[(-1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-26], N[(N[Sin[th], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -3.99999999999999988e-28

   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 99.6%

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

   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. associate-/r/ 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.4%

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

      4. hypot-udef 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. +-commutative 99.4%

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

      8. unpow2 99.4%

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

      9. unpow2 99.4%

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

      10. hypot-def 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in kx around 0 4.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.8%

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

       2. sqrt-unprod 10.0%

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

       3. pow2 10.0%

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

       4. +-commutative 10.0%

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

       5. *-commutative 10.0%

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

       6. fma-def 10.0%

          \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}}^{2}}} \]

   11. Applied egg-rr 10.0%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 10.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right) \cdot \mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)}}} \]

       2. rem-sqrt-square 20.6%

          \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   13. Simplified 20.6%

       \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   if -3.99999999999999988e-28 < (sin.f64 ky) < -2.00000000000000013e-301

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. unpow2 90.4%

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

      3. unpow2 90.4%

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

      4. hypot-def 99.7%

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

   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. associate-/r/ 99.5%

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

      2. div-inv 99.3%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 90.2%

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

      5. unpow2 90.2%

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

      6. unpow2 90.2%

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

      7. +-commutative 90.2%

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

      8. unpow2 90.2%

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

      9. unpow2 90.2%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 40.9%

      \[\leadsto \frac{\color{blue}{\frac{ky}{\sin kx}}}{\frac{1}{\sin th}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 9.5%

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

      2. sqrt-unprod 19.4%

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

      3. pow2 19.4%

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

   8. Applied egg-rr 19.4%

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

      1. unpow2 19.4%

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

      2. rem-sqrt-square 19.2%

         \[\leadsto \frac{\color{blue}{\left|\frac{ky}{\sin kx}\right|}}{\frac{1}{\sin th}} \]

   10. Simplified 19.2%

       \[\leadsto \frac{\color{blue}{\left|\frac{ky}{\sin kx}\right|}}{\frac{1}{\sin th}} \]
   11. Step-by-step derivation

       1. frac-2neg 19.2%

          \[\leadsto \color{blue}{\frac{-\left|\frac{ky}{\sin kx}\right|}{-\frac{1}{\sin th}}} \]

       2. div-inv 19.2%

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

       3. add-sqr-sqrt 9.5%

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

       4. fabs-sqr 9.5%

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

       5. add-sqr-sqrt 40.9%

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

       6. distribute-neg-frac 40.9%

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

       7. distribute-neg-frac 40.9%

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

       8. metadata-eval 40.9%

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

   12. Applied egg-rr 40.9%

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

   if -2.00000000000000013e-301 < (sin.f64 ky) < 5.00000000000000019e-26

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

      2. unpow2 98.0%

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

      3. unpow2 98.0%

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

      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. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 97.8%

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

      5. unpow2 97.8%

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

      6. unpow2 97.8%

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

      7. +-commutative 97.8%

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

      8. unpow2 97.8%

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

      9. unpow2 97.8%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 50.2%

      \[\leadsto \frac{\color{blue}{\frac{ky}{\sin kx}}}{\frac{1}{\sin th}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 60.3%

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

      3. pow2 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. unpow2 60.3%

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

      2. rem-sqrt-square 80.0%

         \[\leadsto \frac{\color{blue}{\left|\frac{ky}{\sin kx}\right|}}{\frac{1}{\sin th}} \]

   10. Simplified 80.0%

       \[\leadsto \frac{\color{blue}{\left|\frac{ky}{\sin kx}\right|}}{\frac{1}{\sin th}} \]

   11. Taylor expanded in ky around 0 80.1%

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

       1. *-commutative 80.1%

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

   13. Simplified 80.1%

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

   if 5.00000000000000019e-26 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 59.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-301}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \frac{-1}{\frac{-1}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 68.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (/ 1.0 (fabs (fma th 0.16666666666666666 (/ 1.0 th))))
   (if (<= (sin ky) 1e-12)
     (* ky (/ (sin th) (hypot (sin ky) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = 1.0 / fabs(fma(th, 0.16666666666666666, (1.0 / th)));
	} else if (sin(ky) <= 1e-12) {
		tmp = ky * (sin(th) / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(1.0 / abs(fma(th, 0.16666666666666666, Float64(1.0 / th))));
	elseif (sin(ky) <= 1e-12)
		tmp = Float64(ky * Float64(sin(th) / hypot(sin(ky), sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(1.0 / N[Abs[N[(th * 0.16666666666666666 + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-12], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   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 99.6%

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

   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. associate-/r/ 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.3%

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

      4. hypot-udef 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. +-commutative 99.3%

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

      8. unpow2 99.3%

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

      9. unpow2 99.3%

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

      10. hypot-def 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in kx around 0 4.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.8%

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

       2. sqrt-unprod 10.4%

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

       3. pow2 10.4%

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

       4. +-commutative 10.4%

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

       5. *-commutative 10.4%

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

       6. fma-def 10.4%

          \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}}^{2}}} \]

   11. Applied egg-rr 10.4%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 10.4%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right) \cdot \mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)}}} \]

       2. rem-sqrt-square 20.2%

          \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   13. Simplified 20.2%

       \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   if -0.029999999999999999 < (sin.f64 ky) < 9.9999999999999998e-13

   1. Initial program 94.3%

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

      1. associate-*l/ 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. hypot-def 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in ky around 0 93.2%

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

      1. expm1-log1p-u 93.2%

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

      2. expm1-udef 31.8%

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

      3. associate-/l* 31.8%

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

   6. Applied egg-rr 31.8%

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

      1. expm1-def 99.0%

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

      2. expm1-log1p 99.0%

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

      3. associate-/r/ 99.2%

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

      4. *-commutative 99.2%

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

   8. Simplified 99.2%

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

   if 9.9999999999999998e-13 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 60.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 68.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (/ 1.0 (fabs (fma th 0.16666666666666666 (/ 1.0 th))))
   (if (<= (sin ky) 1e-12)
     (* (sin th) (/ ky (hypot (sin ky) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = 1.0 / fabs(fma(th, 0.16666666666666666, (1.0 / th)));
	} else if (sin(ky) <= 1e-12) {
		tmp = sin(th) * (ky / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(1.0 / abs(fma(th, 0.16666666666666666, Float64(1.0 / th))));
	elseif (sin(ky) <= 1e-12)
		tmp = Float64(sin(th) * Float64(ky / hypot(sin(ky), sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(1.0 / N[Abs[N[(th * 0.16666666666666666 + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-12], 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 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   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 99.6%

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

   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. associate-/r/ 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.3%

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

      4. hypot-udef 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. +-commutative 99.3%

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

      8. unpow2 99.3%

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

      9. unpow2 99.3%

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

      10. hypot-def 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in kx around 0 4.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.8%

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

       2. sqrt-unprod 10.4%

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

       3. pow2 10.4%

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

       4. +-commutative 10.4%

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

       5. *-commutative 10.4%

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

       6. fma-def 10.4%

          \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}}^{2}}} \]

   11. Applied egg-rr 10.4%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 10.4%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right) \cdot \mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)}}} \]

       2. rem-sqrt-square 20.2%

          \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   13. Simplified 20.2%

       \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   if -0.029999999999999999 < (sin.f64 ky) < 9.9999999999999998e-13

   1. Initial program 94.3%

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

      1. associate-*l/ 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. hypot-def 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in ky around 0 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. times-frac 99.2%

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

      3. /-rgt-identity 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 9.9999999999999998e-13 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 60.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.03)
     (/ (sin ky) (/ t_1 th))
     (if (<= (sin ky) 1e-12) (* (sin th) (/ ky t_1)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = sin(ky) / (t_1 / th);
	} else if (sin(ky) <= 1e-12) {
		tmp = sin(th) * (ky / t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -0.03) {
		tmp = Math.sin(ky) / (t_1 / th);
	} else if (Math.sin(ky) <= 1e-12) {
		tmp = Math.sin(th) * (ky / t_1);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.03:
		tmp = math.sin(ky) / (t_1 / th)
	elif math.sin(ky) <= 1e-12:
		tmp = math.sin(th) * (ky / t_1)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.03)
		tmp = Float64(sin(ky) / Float64(t_1 / th));
	elseif (sin(ky) <= 1e-12)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.03)
		tmp = sin(ky) / (t_1 / th);
	elseif (sin(ky) <= 1e-12)
		tmp = sin(th) * (ky / t_1);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.03], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-12], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. sqr-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. sin-neg 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 50.4%

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

      1. associate-*r/ 50.5%

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

      2. +-commutative 50.5%

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

      3. unpow2 50.5%

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

      4. unpow2 50.5%

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

      5. hypot-def 50.5%

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

      6. *-rgt-identity 50.5%

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

      7. hypot-def 50.5%

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

      8. unpow2 50.5%

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

      9. unpow2 50.5%

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

      10. +-commutative 50.5%

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

      11. unpow2 50.5%

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

      12. unpow2 50.5%

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

      13. hypot-def 50.5%

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

   6. Simplified 50.5%

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

   if -0.029999999999999999 < (sin.f64 ky) < 9.9999999999999998e-13

   1. Initial program 94.3%

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

      1. associate-*l/ 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. hypot-def 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in ky around 0 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. times-frac 99.2%

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

      3. /-rgt-identity 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 9.9999999999999998e-13 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 60.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.03)
   (/
    (/ (sin ky) (hypot (sin kx) (sin ky)))
    (+ (/ 1.0 th) (* th 0.16666666666666666)))
   (if (<= (sin ky) 1e-12)
     (* (sin th) (/ ky (hypot (sin ky) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.03) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (sin(ky) <= 1e-12) {
		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) <= -0.03) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (Math.sin(ky) <= 1e-12) {
		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) <= -0.03:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) / ((1.0 / th) + (th * 0.16666666666666666))
	elif math.sin(ky) <= 1e-12:
		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) <= -0.03)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	elseif (sin(ky) <= 1e-12)
		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) <= -0.03)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) / ((1.0 / th) + (th * 0.16666666666666666));
	elseif (sin(ky) <= 1e-12)
		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], -0.03], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-12], 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 3 regimes

2. if (sin.f64 ky) < -0.029999999999999999

   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 99.6%

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

   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. associate-/r/ 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.3%

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

      4. hypot-udef 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. +-commutative 99.3%

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

      8. unpow2 99.3%

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

      9. unpow2 99.3%

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

      10. hypot-def 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   if -0.029999999999999999 < (sin.f64 ky) < 9.9999999999999998e-13

   1. Initial program 94.3%

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

      1. associate-*l/ 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. hypot-def 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in ky around 0 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. times-frac 99.2%

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

      3. /-rgt-identity 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 9.9999999999999998e-13 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 60.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.03:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 43.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -4e-28)
   (/ 1.0 (fabs (fma th 0.16666666666666666 (/ 1.0 th))))
   (if (<= (sin ky) 5e-26) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -4e-28) {
		tmp = 1.0 / fabs(fma(th, 0.16666666666666666, (1.0 / th)));
	} else if (sin(ky) <= 5e-26) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -4e-28)
		tmp = Float64(1.0 / abs(fma(th, 0.16666666666666666, Float64(1.0 / th))));
	elseif (sin(ky) <= 5e-26)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -4e-28], N[(1.0 / N[Abs[N[(th * 0.16666666666666666 + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-26], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -3.99999999999999988e-28

   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 99.6%

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

   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. associate-/r/ 99.6%

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

      2. div-inv 99.6%

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

      3. associate-/r* 99.4%

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

      4. hypot-udef 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. +-commutative 99.4%

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

      8. unpow2 99.4%

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

      9. unpow2 99.4%

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

      10. hypot-def 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in kx around 0 4.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.8%

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

       2. sqrt-unprod 10.0%

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

       3. pow2 10.0%

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

       4. +-commutative 10.0%

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

       5. *-commutative 10.0%

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

       6. fma-def 10.0%

          \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}}^{2}}} \]

   11. Applied egg-rr 10.0%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right)}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 10.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right) \cdot \mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)}}} \]

       2. rem-sqrt-square 20.6%

          \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   13. Simplified 20.6%

       \[\leadsto \frac{1}{\color{blue}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}} \]

   if -3.99999999999999988e-28 < (sin.f64 ky) < 5.00000000000000019e-26

   1. Initial program 94.0%

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

   2. Taylor expanded in ky around 0 45.4%

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

   if 5.00000000000000019e-26 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 59.8%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\left|\mathsf{fma}\left(th, 0.16666666666666666, \frac{1}{th}\right)\right|}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 34.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-52)
   (* (sin ky) (/ th (sin kx)))
   (if (<= (sin ky) 1e-143) (/ (/ ky kx) (/ 1.0 (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-52) {
		tmp = sin(ky) * (th / sin(kx));
	} else if (sin(ky) <= 1e-143) {
		tmp = (ky / kx) / (1.0 / sin(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 (sin(ky) <= (-2d-52)) then
        tmp = sin(ky) * (th / sin(kx))
    else if (sin(ky) <= 1d-143) then
        tmp = (ky / kx) / (1.0d0 / sin(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 (Math.sin(ky) <= -2e-52) {
		tmp = Math.sin(ky) * (th / Math.sin(kx));
	} else if (Math.sin(ky) <= 1e-143) {
		tmp = (ky / kx) / (1.0 / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-52:
		tmp = math.sin(ky) * (th / math.sin(kx))
	elif math.sin(ky) <= 1e-143:
		tmp = (ky / kx) / (1.0 / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-52)
		tmp = Float64(sin(ky) * Float64(th / sin(kx)));
	elseif (sin(ky) <= 1e-143)
		tmp = Float64(Float64(ky / kx) / Float64(1.0 / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-52)
		tmp = sin(ky) * (th / sin(kx));
	elseif (sin(ky) <= 1e-143)
		tmp = (ky / kx) / (1.0 / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-52], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-143], N[(N[(ky / kx), $MachinePrecision] / N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2e-52

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. sqr-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. sin-neg 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 10.2%

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

   5. Taylor expanded in th around 0 9.6%

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

      1. associate-/l* 9.7%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in ky around inf 9.6%

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

      1. associate-*r/ 9.7%

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

   10. Simplified 9.7%

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

   if -2e-52 < (sin.f64 ky) < 9.9999999999999995e-144

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. unpow2 91.5%

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

      3. unpow2 91.5%

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

      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. associate-/r/ 99.6%

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

      2. div-inv 99.4%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 91.3%

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

      5. unpow2 91.3%

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

      6. unpow2 91.3%

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

      7. +-commutative 91.3%

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

      8. unpow2 91.3%

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

      9. unpow2 91.3%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 50.5%

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

   7. Taylor expanded in kx around 0 35.6%

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

   if 9.9999999999999995e-144 < (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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.3%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 34.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-143) (* (sin ky) (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-143) {
		tmp = sin(ky) * (sin(th) / 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) <= 1d-143) then
        tmp = sin(ky) * (sin(th) / 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) <= 1e-143) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-143:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-143)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-143)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-143], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999995e-144

   1. Initial program 95.1%

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

      1. associate-/r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. unpow2 95.0%

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

      4. sqr-neg 95.0%

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

      5. sin-neg 95.0%

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

      6. sin-neg 95.0%

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

      7. unpow2 95.0%

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

      8. +-commutative 95.0%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 32.5%

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

      1. *-un-lft-identity 32.5%

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

      2. div-inv 32.4%

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

      3. times-frac 31.6%

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

   6. Applied egg-rr 31.6%

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

   7. Taylor expanded in kx around 0 21.0%

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

      1. *-commutative 21.0%

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

      2. *-lft-identity 21.0%

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

      3. times-frac 21.8%

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

      4. /-rgt-identity 21.8%

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

   9. Simplified 21.8%

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

   if 9.9999999999999995e-144 < (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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.3%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 39.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-26) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-26) {
		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) <= 5d-26) 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) <= 5e-26) {
		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) <= 5e-26:
		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) <= 5e-26)
		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) <= 5e-26)
		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], 5e-26], 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) < 5.00000000000000019e-26

   1. Initial program 95.9%

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

   2. Taylor expanded in ky around 0 31.1%

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

   if 5.00000000000000019e-26 < (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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 59.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 33.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-143) (/ (/ ky kx) (/ 1.0 (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-143) {
		tmp = (ky / kx) / (1.0 / sin(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 (sin(ky) <= 1d-143) then
        tmp = (ky / kx) / (1.0d0 / sin(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 (Math.sin(ky) <= 1e-143) {
		tmp = (ky / kx) / (1.0 / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-143:
		tmp = (ky / kx) / (1.0 / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-143)
		tmp = Float64(Float64(ky / kx) / Float64(1.0 / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-143)
		tmp = (ky / kx) / (1.0 / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-143], N[(N[(ky / kx), $MachinePrecision] / N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999995e-144

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

      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. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 94.9%

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

      5. unpow2 94.9%

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

      6. unpow2 94.9%

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

      7. +-commutative 94.9%

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

      8. unpow2 94.9%

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

      9. unpow2 94.9%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 30.5%

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

   7. Taylor expanded in kx around 0 21.3%

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

   if 9.9999999999999995e-144 < (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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.3%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 33.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-143) (/ ky (/ (sin kx) th)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-143) {
		tmp = ky / (sin(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 (sin(ky) <= 1d-143) then
        tmp = ky / (sin(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 (Math.sin(ky) <= 1e-143) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-143:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-143)
		tmp = Float64(ky / Float64(sin(kx) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-143)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-143], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999995e-144

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

      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. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 94.9%

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

      5. unpow2 94.9%

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

      6. unpow2 94.9%

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

      7. +-commutative 94.9%

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

      8. unpow2 94.9%

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

      9. unpow2 94.9%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 30.5%

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

   7. Taylor expanded in th around 0 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/l* 20.2%

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

   9. Simplified 20.2%

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

   if 9.9999999999999995e-144 < (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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.3%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 33.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-143) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-143) {
		tmp = sin(th) / (kx / ky);
	} 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) <= 1d-143) then
        tmp = sin(th) / (kx / ky)
    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) <= 1e-143) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-143:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-143)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-143)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-143], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999995e-144

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

      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. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.5%

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

      4. hypot-udef 94.9%

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

      5. unpow2 94.9%

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

      6. unpow2 94.9%

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

      7. +-commutative 94.9%

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

      8. unpow2 94.9%

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

      9. unpow2 94.9%

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

      10. hypot-def 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in ky around 0 30.5%

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

   7. Taylor expanded in kx around 0 20.6%

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

      1. associate-/l* 21.3%

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

   9. Simplified 21.3%

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

   if 9.9999999999999995e-144 < (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 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.3%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-143}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 24.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

   1. +-commutative 97.0%

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

   2. unpow2 97.0%

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

   3. unpow2 97.0%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in kx around 0 24.5%

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

5. Final simplification 24.5%

   \[\leadsto \sin th \]

Alternative 15: 14.5% accurate, 78.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

   1. +-commutative 97.0%

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

   2. unpow2 97.0%

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

   3. unpow2 97.0%

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

   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. associate-/r/ 99.6%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.5%

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

   4. hypot-udef 96.8%

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

   5. unpow2 96.8%

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

   6. unpow2 96.8%

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

   7. +-commutative 96.8%

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

   8. unpow2 96.8%

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

   9. unpow2 96.8%

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

   10. hypot-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in th around 0 49.3%

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

   1. *-commutative 49.3%

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

8. Simplified 49.3%

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

9. Taylor expanded in kx around 0 14.2%

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

10. Final simplification 14.2%

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

Alternative 16: 13.8% 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 97.0%

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

   1. +-commutative 97.0%

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

   2. unpow2 97.0%

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

   3. unpow2 97.0%

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

   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. associate-/r/ 99.6%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.5%

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

   4. hypot-udef 96.8%

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

   5. unpow2 96.8%

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

   6. unpow2 96.8%

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

   7. +-commutative 96.8%

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

   8. unpow2 96.8%

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

   9. unpow2 96.8%

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

   10. hypot-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in th around 0 49.3%

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

   1. *-commutative 49.3%

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

8. Simplified 49.3%

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

9. Taylor expanded in kx around 0 14.2%

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

10. Taylor expanded in th around 0 13.6%

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

11. Final simplification 13.6%

    \[\leadsto th \]

Reproduce

herbie shell --seed 2023274 
(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)))