Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 32.4s

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 94.9%

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

2. Simplified 99.7%

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

   [Start] 94.9	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 94.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 94.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 94.9	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 99.7	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Applied egg-rr 99.7%

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

   [Start] 99.7	\[ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]
   *-commutative [=>] 99.7	\[ \color{blue}{\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   clear-num [=>] 99.6	\[ \sin th \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]
   un-div-inv [=>] 99.7	\[ \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	60.4%
Cost	58644

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin th \leq -0.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.71:\\ \;\;\;\;\sin ky \cdot \frac{t_1}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.4235:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.0001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Accuracy	60.4%
Cost	58644

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin th \leq -0.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.71:\\ \;\;\;\;\sin ky \cdot \frac{t_1}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.4235:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.0001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 4 \cdot 10^{-7}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Accuracy	60.4%
Cost	58644

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin th \leq -0.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq -0.71:\\ \;\;\;\;\sin ky \cdot \frac{t_1}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.4235:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.0001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Accuracy	45.2%
Cost	45648

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\sin ky \cdot \frac{t_1}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -8 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 5
Accuracy	76.5%
Cost	39432

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Accuracy	76.3%
Cost	39176

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	39176

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Accuracy	46.2%
Cost	32780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-124}:\\ \;\;\;\;ky \cdot \frac{th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Accuracy	99.6%
Cost	32384

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

Alternative 10
Accuracy	99.7%
Cost	32384

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

Alternative 11
Accuracy	74.9%
Cost	26249

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -3.4 \cdot 10^{-6} \lor \neg \left(th \leq 7.2\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \end{array} \]

Alternative 12
Accuracy	47.6%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Accuracy	34.8%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sin ky}{\frac{ky}{-th}}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Accuracy	34.9%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{th \cdot \left(-\sin ky\right)}{ky}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Accuracy	41.7%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Accuracy	32.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Accuracy	33.5%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Accuracy	33.5%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19
Accuracy	21.7%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.5 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \mathbf{elif}\;ky \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \end{array} \]

Alternative 20
Accuracy	30.7%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -6 \cdot 10^{-35}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{-145}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 21
Accuracy	20.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 22
Accuracy	13.3%
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023163 
(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)))