Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 18.3s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% 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 91.0%

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

   1. associate-*l/ 88.4%

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

   2. *-commutative 88.4%

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

   3. associate-*l/ 90.9%

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

   4. remove-double-neg 90.9%

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

   5. sin-neg 90.9%

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

   6. distribute-rgt-neg-in 90.9%

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

   7. associate-/r/ 90.9%

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

   8. distribute-neg-frac 90.9%

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

   9. neg-mul-1 90.9%

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

   10. associate-*l/ 90.9%

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

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: 48.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191} \lor \neg \left(\sin kx \leq 10^{-182}\right) \land \sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (* ky (- (sin th))) (sin kx))
   (if (or (<= (sin kx) 5e-191)
           (and (not (<= (sin kx) 1e-182)) (<= (sin kx) 5e-92)))
     (sin th)
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if ((sin(kx) <= 5e-191) || (!(sin(kx) <= 1e-182) && (sin(kx) <= 5e-92))) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = (ky * -sin(th)) / sin(kx)
    else if ((sin(kx) <= 5d-191) .or. (.not. (sin(kx) <= 1d-182)) .and. (sin(kx) <= 5d-92)) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if ((Math.sin(kx) <= 5e-191) || (!(Math.sin(kx) <= 1e-182) && (Math.sin(kx) <= 5e-92))) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif (math.sin(kx) <= 5e-191) or (not (math.sin(kx) <= 1e-182) and (math.sin(kx) <= 5e-92)):
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif ((sin(kx) <= 5e-191) || (!(sin(kx) <= 1e-182) && (sin(kx) <= 5e-92)))
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif ((sin(kx) <= 5e-191) || (~((sin(kx) <= 1e-182)) && (sin(kx) <= 5e-92)))
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], And[N[Not[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182]], $MachinePrecision], LessEqual[N[Sin[kx], $MachinePrecision], 5e-92]]], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

      4. distribute-rgt-neg-in 61.0%

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

   9. Simplified 61.0%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182 or 5.00000000000000011e-92 < (sin.f64 kx)

   1. Initial program 94.8%

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

      1. frac-2neg 94.8%

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

      2. +-commutative 94.8%

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

      3. unpow2 94.8%

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

      4. unpow2 94.8%

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

      5. hypot-udef 99.6%

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

      6. frac-2neg 99.6%

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

      7. expm1-log1p-u 99.5%

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

      8. expm1-udef 40.4%

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

   3. Applied egg-rr 40.4%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in ky around 0 62.4%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191} \lor \neg \left(\sin kx \leq 10^{-182}\right) \land \sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 3: 48.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 5e-191)
     (sin th)
     (if (<= (sin kx) 1e-182)
       (* (sin ky) (/ (sin th) (sin kx)))
       (if (<= (sin kx) 5e-92) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 5e-191) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-182) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(kx) <= 5e-92) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = (ky * -sin(th)) / sin(kx)
    else if (sin(kx) <= 5d-191) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-182) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else if (sin(kx) <= 5d-92) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 5e-191) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-182) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(kx) <= 5e-92) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 5e-191:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-182:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(kx) <= 5e-92:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-92], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

      4. distribute-rgt-neg-in 61.0%

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

   9. Simplified 61.0%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182

   1. Initial program 2.2%

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

      1. frac-2neg 2.2%

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

      2. +-commutative 2.2%

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

      3. unpow2 2.2%

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

      4. unpow2 2.2%

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

      5. hypot-udef 99.2%

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

      6. frac-2neg 99.2%

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

      7. expm1-log1p-u 99.2%

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

      8. expm1-udef 6.8%

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

   3. Applied egg-rr 6.8%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in ky around 0 100.0%

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

   if 5.00000000000000011e-92 < (sin.f64 kx)

   1. Initial program 99.6%

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

   2. Taylor expanded in ky around 0 60.4%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 4: 45.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{1}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 5e-191)
     (sin th)
     (if (<= (sin kx) 1e-182)
       (* (sin ky) (/ 1.0 (/ kx (sin th))))
       (if (<= (sin kx) 5e-92)
         (sin th)
         (/
          (/ (sin th) (sin kx))
          (+ (* ky 0.16666666666666666) (/ 1.0 ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 5e-191) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-182) {
		tmp = sin(ky) * (1.0 / (kx / sin(th)));
	} else if (sin(kx) <= 5e-92) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / sin(kx)) / ((ky * 0.16666666666666666) + (1.0 / ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = (ky * -sin(th)) / sin(kx)
    else if (sin(kx) <= 5d-191) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-182) then
        tmp = sin(ky) * (1.0d0 / (kx / sin(th)))
    else if (sin(kx) <= 5d-92) then
        tmp = sin(th)
    else
        tmp = (sin(th) / sin(kx)) / ((ky * 0.16666666666666666d0) + (1.0d0 / ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 5e-191) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-182) {
		tmp = Math.sin(ky) * (1.0 / (kx / Math.sin(th)));
	} else if (Math.sin(kx) <= 5e-92) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) / Math.sin(kx)) / ((ky * 0.16666666666666666) + (1.0 / ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 5e-191:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-182:
		tmp = math.sin(ky) * (1.0 / (kx / math.sin(th)))
	elif math.sin(kx) <= 5e-92:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) / math.sin(kx)) / ((ky * 0.16666666666666666) + (1.0 / ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = Float64(sin(ky) * Float64(1.0 / Float64(kx / sin(th))));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / sin(kx)) / Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = sin(ky) * (1.0 / (kx / sin(th)));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = (sin(th) / sin(kx)) / ((ky * 0.16666666666666666) + (1.0 / ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182], N[(N[Sin[ky], $MachinePrecision] * N[(1.0 / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-92], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] / N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

      4. distribute-rgt-neg-in 61.0%

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

   9. Simplified 61.0%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182

   1. Initial program 2.2%

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

      1. associate-*l/ 1.4%

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

      2. +-commutative 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-udef 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-/l* 98.8%

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

      8. div-inv 98.8%

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

      9. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in ky around 0 99.6%

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

   5. Taylor expanded in kx around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 5.00000000000000011e-92 < (sin.f64 kx)

   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-*l/ 97.4%

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

      2. +-commutative 97.4%

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

      3. unpow2 97.4%

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

      4. unpow2 97.4%

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

      5. hypot-udef 97.5%

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

      6. *-commutative 97.5%

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

      7. associate-/l* 99.4%

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

      8. div-inv 99.3%

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

      9. associate-/r* 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in ky around 0 60.4%

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

   5. Taylor expanded in ky around 0 53.6%

      \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{0.16666666666666666 \cdot ky + \frac{1}{ky}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{1}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \end{array} \]

Alternative 5: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (sqrt (pow (sin th) 2.0))
   (if (<= (sin ky) 0.04)
     (/
      (/ (sin th) (hypot (sin kx) (sin ky)))
      (+ (* ky 0.16666666666666666) (/ 1.0 ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 0.04) {
		tmp = (sin(th) / hypot(sin(kx), sin(ky))) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 0.04) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.sin(ky))) / ((ky * 0.16666666666666666) + (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 0.04:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.sin(ky))) / ((ky * 0.16666666666666666) + (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 0.04)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), sin(ky))) / Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 0.04)
		tmp = (sin(th) / hypot(sin(kx), sin(ky))) / ((ky * 0.16666666666666666) + (1.0 / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.04], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. frac-2neg 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. frac-2neg 99.8%

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

      7. expm1-log1p-u 99.6%

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

      8. expm1-udef 55.6%

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

   3. Applied egg-rr 55.4%

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

      1. expm1-def 99.4%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.6%

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

      4. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 0.7%

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

      5. sqrt-unprod 36.2%

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

      6. pow2 36.2%

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

   8. Applied egg-rr 36.2%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 0.0400000000000000008

   1. Initial program 84.1%

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

      1. associate-*l/ 79.5%

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

      2. +-commutative 79.5%

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

      3. unpow2 79.5%

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

      4. unpow2 79.5%

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

      5. hypot-udef 91.3%

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

      6. *-commutative 91.3%

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

      7. associate-/l* 99.6%

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

      8. div-inv 99.5%

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

      9. associate-/r* 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in ky around 0 97.3%

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

   if 0.0400000000000000008 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 68.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 45.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{-ky}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (* (sin th) (/ (- ky) (sin kx)))
   (if (<= (sin kx) 5e-191)
     (sin th)
     (if (<= (sin kx) 1e-182)
       (* (sin th) (/ ky kx))
       (if (<= (sin kx) 5e-92) (sin th) (* (sin th) (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = sin(th) * (-ky / sin(kx));
	} else if (sin(kx) <= 5e-191) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-182) {
		tmp = sin(th) * (ky / kx);
	} else if (sin(kx) <= 5e-92) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = sin(th) * (-ky / sin(kx))
    else if (sin(kx) <= 5d-191) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-182) then
        tmp = sin(th) * (ky / kx)
    else if (sin(kx) <= 5d-92) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = Math.sin(th) * (-ky / Math.sin(kx));
	} else if (Math.sin(kx) <= 5e-191) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-182) {
		tmp = Math.sin(th) * (ky / kx);
	} else if (Math.sin(kx) <= 5e-92) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = math.sin(th) * (-ky / math.sin(kx))
	elif math.sin(kx) <= 5e-191:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-182:
		tmp = math.sin(th) * (ky / kx)
	elif math.sin(kx) <= 5e-92:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(sin(th) * Float64(Float64(-ky) / sin(kx)));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = Float64(sin(th) * Float64(ky / kx));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = sin(th) * (-ky / sin(kx));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = sin(th) * (ky / kx);
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[Sin[th], $MachinePrecision] * N[((-ky) / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-92], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.1%

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

      1. associate-*r/ 61.1%

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

      2. neg-mul-1 61.1%

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

   9. Simplified 61.1%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182

   1. Initial program 2.2%

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

   2. Taylor expanded in ky around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in kx around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.2%

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

   7. Simplified 99.2%

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

   if 5.00000000000000011e-92 < (sin.f64 kx)

   1. Initial program 99.6%

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

   2. Taylor expanded in ky around 0 50.7%

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

      1. associate-/l* 52.8%

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

      2. associate-/r/ 52.8%

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

   4. Simplified 52.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{-ky}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 7: 45.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 5e-191)
     (sin th)
     (if (<= (sin kx) 1e-182)
       (* (sin th) (/ ky kx))
       (if (<= (sin kx) 5e-92) (sin th) (* (sin th) (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 5e-191) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-182) {
		tmp = sin(th) * (ky / kx);
	} else if (sin(kx) <= 5e-92) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = (ky * -sin(th)) / sin(kx)
    else if (sin(kx) <= 5d-191) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-182) then
        tmp = sin(th) * (ky / kx)
    else if (sin(kx) <= 5d-92) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 5e-191) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-182) {
		tmp = Math.sin(th) * (ky / kx);
	} else if (Math.sin(kx) <= 5e-92) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 5e-191:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-182:
		tmp = math.sin(th) * (ky / kx)
	elif math.sin(kx) <= 5e-92:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = Float64(sin(th) * Float64(ky / kx));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = sin(th) * (ky / kx);
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-92], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

      4. distribute-rgt-neg-in 61.0%

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

   9. Simplified 61.0%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182

   1. Initial program 2.2%

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

   2. Taylor expanded in ky around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in kx around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.2%

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

   7. Simplified 99.2%

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

   if 5.00000000000000011e-92 < (sin.f64 kx)

   1. Initial program 99.6%

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

   2. Taylor expanded in ky around 0 50.7%

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

      1. associate-/l* 52.8%

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

      2. associate-/r/ 52.8%

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

   4. Simplified 52.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 8: 45.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{1}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (/ (* ky (- (sin th))) (sin kx))
   (if (<= (sin kx) 5e-191)
     (sin th)
     (if (<= (sin kx) 1e-182)
       (* (sin ky) (/ 1.0 (/ kx (sin th))))
       (if (<= (sin kx) 5e-92) (sin th) (* (sin th) (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (ky * -sin(th)) / sin(kx);
	} else if (sin(kx) <= 5e-191) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-182) {
		tmp = sin(ky) * (1.0 / (kx / sin(th)));
	} else if (sin(kx) <= 5e-92) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = (ky * -sin(th)) / sin(kx)
    else if (sin(kx) <= 5d-191) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-182) then
        tmp = sin(ky) * (1.0d0 / (kx / sin(th)))
    else if (sin(kx) <= 5d-92) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (ky * -Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(kx) <= 5e-191) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-182) {
		tmp = Math.sin(ky) * (1.0 / (kx / Math.sin(th)));
	} else if (Math.sin(kx) <= 5e-92) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (ky * -math.sin(th)) / math.sin(kx)
	elif math.sin(kx) <= 5e-191:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-182:
		tmp = math.sin(ky) * (1.0 / (kx / math.sin(th)))
	elif math.sin(kx) <= 5e-92:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(ky * Float64(-sin(th))) / sin(kx));
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = Float64(sin(ky) * Float64(1.0 / Float64(kx / sin(th))));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (ky * -sin(th)) / sin(kx);
	elseif (sin(kx) <= 5e-191)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-182)
		tmp = sin(ky) * (1.0 / (kx / sin(th)));
	elseif (sin(kx) <= 5e-92)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(ky * (-N[Sin[th], $MachinePrecision])), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-191], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-182], N[(N[Sin[ky], $MachinePrecision] * N[(1.0 / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-92], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 22.3%

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

      2. associate-/r/ 22.3%

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

   4. Simplified 22.3%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 33.1%

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

      3. pow1 33.1%

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

      4. pow-plus 33.1%

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

      5. metadata-eval 33.1%

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

   6. Applied egg-rr 33.1%

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

   7. Taylor expanded in ky around -inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

      3. *-commutative 61.0%

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

      4. distribute-rgt-neg-in 61.0%

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

   9. Simplified 61.0%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 5.0000000000000001e-191 or 1e-182 < (sin.f64 kx) < 5.00000000000000011e-92

   1. Initial program 84.1%

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

   2. Taylor expanded in kx around 0 46.7%

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

   if 5.0000000000000001e-191 < (sin.f64 kx) < 1e-182

   1. Initial program 2.2%

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

      1. associate-*l/ 1.4%

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

      2. +-commutative 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-udef 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-/l* 98.8%

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

      8. div-inv 98.8%

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

      9. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in ky around 0 99.6%

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

   5. Taylor expanded in kx around 0 55.0%

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

      1. associate-/l* 99.6%

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

      2. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 5.00000000000000011e-92 < (sin.f64 kx)

   1. Initial program 99.6%

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

   2. Taylor expanded in ky around 0 50.7%

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

      1. associate-/l* 52.8%

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

      2. associate-/r/ 52.8%

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

   4. Simplified 52.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{ky \cdot \left(-\sin th\right)}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-182}:\\ \;\;\;\;\sin ky \cdot \frac{1}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 9: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (sqrt (pow (sin th) 2.0))
   (if (<= (sin ky) 5e-7)
     (/ (/ (sin th) (hypot (sin kx) (sin ky))) (/ 1.0 ky))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 5e-7) {
		tmp = (sin(th) / hypot(sin(kx), sin(ky))) / (1.0 / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.002) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 5e-7) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.sin(ky))) / (1.0 / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.002:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 5e-7:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.sin(ky))) / (1.0 / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-7)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), sin(ky))) / Float64(1.0 / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.002)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 5e-7)
		tmp = (sin(th) / hypot(sin(kx), sin(ky))) / (1.0 / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-7], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(1.0 / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2e-3

   1. Initial program 99.7%

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

      1. frac-2neg 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. frac-2neg 99.8%

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

      7. expm1-log1p-u 99.6%

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

      8. expm1-udef 55.3%

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

   3. Applied egg-rr 55.2%

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

      1. expm1-def 99.4%

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

      2. expm1-log1p 99.5%

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

      3. associate-*r/ 99.6%

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

      4. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 0.7%

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

      5. sqrt-unprod 35.2%

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

      6. pow2 35.2%

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

   8. Applied egg-rr 35.2%

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

   if -2e-3 < (sin.f64 ky) < 4.99999999999999977e-7

   1. Initial program 83.7%

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

      1. associate-*l/ 78.9%

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

      2. +-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. unpow2 78.9%

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

      5. hypot-udef 91.1%

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

      6. *-commutative 91.1%

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

      7. associate-/l* 99.6%

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

      8. div-inv 99.5%

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

      9. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in ky around 0 99.2%

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

   if 4.99999999999999977e-7 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 66.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. frac-2neg 91.0%

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

   2. +-commutative 91.0%

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

   3. unpow2 91.0%

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

   4. unpow2 91.0%

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

   5. hypot-udef 99.7%

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

   6. frac-2neg 99.7%

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

   7. expm1-log1p-u 99.6%

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

   8. expm1-udef 49.1%

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

3. Applied egg-rr 49.0%

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

   1. expm1-def 99.5%

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

   2. expm1-log1p 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 11: 33.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.9 \cdot 10^{-22}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.9e-22) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.9e-22) {
		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 (ky <= 3.9d-22) 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 (ky <= 3.9e-22) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.9e-22:
		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 (ky <= 3.9e-22)
		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 (ky <= 3.9e-22)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.9e-22], 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 ky < 3.89999999999999998e-22

   1. Initial program 87.8%

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

   2. Taylor expanded in ky around 0 33.7%

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

      1. associate-/l* 35.4%

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

      2. associate-/r/ 35.4%

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

   4. Simplified 35.4%

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

   if 3.89999999999999998e-22 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 40.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.9 \cdot 10^{-22}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 27.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6e-141) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-141) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 6d-141) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-141) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 6e-141:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 6e-141)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 6e-141)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 6e-141], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.99999999999999967e-141

   1. Initial program 85.7%

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

   2. Taylor expanded in ky around 0 31.4%

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

      1. associate-/l* 33.5%

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

      2. associate-/r/ 33.4%

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

   4. Simplified 33.4%

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

   5. Taylor expanded in kx around 0 21.4%

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

      1. associate-/l* 23.5%

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

      2. associate-/r/ 23.4%

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

   7. Simplified 23.4%

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

   if 5.99999999999999967e-141 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.1%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 27.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.8e-142) (/ ky (/ kx (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.8e-142) {
		tmp = ky / (kx / 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 (ky <= 2.8d-142) then
        tmp = ky / (kx / 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 (ky <= 2.8e-142) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.8e-142:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.8e-142)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.8e-142)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.8e-142], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.80000000000000004e-142

   1. Initial program 85.7%

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

   2. Taylor expanded in ky around 0 31.4%

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

      1. associate-/l* 33.5%

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

      2. associate-/r/ 33.4%

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

   4. Simplified 33.4%

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

   5. Taylor expanded in kx around 0 21.4%

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

      1. associate-/l* 23.5%

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

   7. Simplified 23.5%

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

   if 2.80000000000000004e-142 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.1%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 24.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.15 \cdot 10^{-143}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.15e-143) (* th (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.15e-143) {
		tmp = th * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3.15d-143) then
        tmp = th * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.15e-143) {
		tmp = th * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.15e-143:
		tmp = th * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.15e-143)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.15e-143)
		tmp = th * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.15e-143], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 3.15000000000000019e-143

   1. Initial program 85.7%

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

   2. Taylor expanded in ky around 0 31.4%

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

      1. associate-/l* 33.5%

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

      2. associate-/r/ 33.4%

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

   4. Simplified 33.4%

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

   5. Taylor expanded in kx around 0 21.4%

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

      1. associate-/l* 23.5%

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

      2. associate-/r/ 23.4%

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

   7. Simplified 23.4%

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

   8. Taylor expanded in th around 0 17.0%

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

      1. associate-/l* 19.0%

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

      2. associate-/r/ 19.1%

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

   10. Simplified 19.1%

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

   if 3.15000000000000019e-143 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 41.1%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.15 \cdot 10^{-143}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 18.1% accurate, 100.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.6e-52) (* th (/ ky kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.6e-52) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3.6d-52) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.6e-52) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.6e-52:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.6e-52)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.6e-52)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.6e-52], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 3.59999999999999988e-52

   1. Initial program 87.3%

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

   2. Taylor expanded in ky around 0 34.3%

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

      1. associate-/l* 36.1%

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

      2. associate-/r/ 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in kx around 0 22.5%

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

      1. associate-/l* 24.3%

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

      2. associate-/r/ 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in th around 0 18.4%

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

      1. associate-/l* 20.2%

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

      2. associate-/r/ 20.3%

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

   10. Simplified 20.3%

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

   if 3.59999999999999988e-52 < 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. frac-2neg 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. frac-2neg 99.8%

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

      7. expm1-log1p-u 99.6%

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

      8. expm1-udef 56.7%

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

   3. Applied egg-rr 56.5%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.7%

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

      4. associate-/l* 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in kx around 0 40.8%

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

   7. Taylor expanded in th around 0 16.3%

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

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16: 14.3% 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 91.0%

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

   1. frac-2neg 91.0%

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

   2. +-commutative 91.0%

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

   3. unpow2 91.0%

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

   4. unpow2 91.0%

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

   5. hypot-udef 99.7%

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

   6. frac-2neg 99.7%

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

   7. expm1-log1p-u 99.6%

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

   8. expm1-udef 49.1%

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

3. Applied egg-rr 49.0%

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

   1. expm1-def 99.5%

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

   2. expm1-log1p 99.6%

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

   3. associate-*r/ 95.0%

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

   4. associate-/l* 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in kx around 0 26.7%

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

7. Taylor expanded in th around 0 13.1%

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

8. Final simplification 13.1%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023336 
(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)))