Toniolo and Linder, Equation (3b), real

Average Accuracy: 94.1% → 99.6%

Time: 47.0s

Precision: binary64

Cost: 32384

Math

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

↓

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

FPCore

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

↓

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

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);
}

↓

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

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);
}

↓

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

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)

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 94.1%

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

2. Simplified 99.7%

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

   [Start] 94.1	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 94.1	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 94.1	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 94.1	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 99.7	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Applied egg-rr 43.5%

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

4. Simplified 99.6%

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

   [Start] 43.5	\[ e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]
   expm1-def [=>] 99.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]
   expm1-log1p [=>] 99.6	\[ \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   *-commutative [=>] 99.6	\[ \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   associate-/r/ [<=] 99.6	\[ \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

5. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	74.0%
Cost	52496

\[\begin{array}{l} t_1 := \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{if}\;\sin th \leq -0.61:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.25:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin th \leq -0.01:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq 0.05:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	76.4%
Cost	52112

\[\begin{array}{l} t_1 := \frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{if}\;\sin kx \leq -0.78:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -0.4415:\\ \;\;\;\;\frac{\left|\sin th \cdot \sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq -0.002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 3
Accuracy	76.4%
Cost	52112

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;\sin kx \leq -0.78:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq -0.4415:\\ \;\;\;\;\frac{\left|\sin th \cdot \sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 4
Accuracy	76.4%
Cost	52112

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.78:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq -0.4415:\\ \;\;\;\;\frac{\left|\sin th \cdot \sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq -0.002:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 5
Accuracy	45.3%
Cost	39116

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.165:\\ \;\;\;\;\frac{\left|\sin th \cdot \sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 6
Accuracy	73.3%
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.165:\\ \;\;\;\;\frac{\left|\sin th \cdot \sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 7
Accuracy	40.9%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 8
Accuracy	40.9%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 9
Accuracy	42.3%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 10
Accuracy	45.3%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 11
Accuracy	99.7%
Cost	32384

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

Alternative 12
Accuracy	73.9%
Cost	26380

\[\begin{array}{l} t_1 := \frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;th \leq -4.3 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 24000000:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;th \leq 7.2 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 13
Accuracy	32.6%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 14
Accuracy	34.3%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sin th}{ky \cdot \left(0.5 \cdot \frac{1}{kx} + kx \cdot 0.16666666666666666\right) + \frac{kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Accuracy	39.4%
Cost	26184

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-133}:\\ \;\;\;\;\sin th \cdot t_1\\ \mathbf{elif}\;\sin ky \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Accuracy	39.4%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-133}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Accuracy	30.6%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.002:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 18
Accuracy	31.2%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \]

Alternative 19
Accuracy	34.3%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin th}{ky \cdot \left(0.5 \cdot \frac{1}{kx} + kx \cdot 0.16666666666666666\right) + \frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20
Accuracy	34.2%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky} + 0.16666666666666666 \cdot \left(ky \cdot kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 21
Accuracy	30.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.8 \cdot 10^{+63}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;kx \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \end{array} \]

Alternative 22
Accuracy	30.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;kx \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;kx \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \end{array} \]

Alternative 23
Accuracy	31.0%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.75 \cdot 10^{-53}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.5 \cdot 10^{-184}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 24
Accuracy	21.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -5.2 \cdot 10^{-54}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 25
Accuracy	13.7%
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023129 
(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)))