Toniolo and Linder, Equation (3b), real

Average Error: 93.7% → 99.7%

Time: 25.8s

Precision: binary64

Cost: 32384.00

Math

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

↓

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

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 ky) (hypot (sin ky) (sin kx))) (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);
}

↓

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.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(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * 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)

↓

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) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

↓

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) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

↓

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[(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[(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 93.7

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

2. Simplified 99.7

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

   [Start] 93.7	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 93.7	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 93.7	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 93.7	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 99.7	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Final simplification 99.7

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

Alternatives

Alternative 1
Error	48.8%
Cost	45516.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-140}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	45.1%
Cost	39308.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-140}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky}}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	70.0%
Cost	39048.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	77.0%
Cost	39048.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0001:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	44.4%
Cost	32712.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	41.4%
Cost	26184.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.0001:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	44.4%
Cost	26184.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-191}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	34.4%
Cost	19784.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-191}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	36.0%
Cost	19784.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-191}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	29.1%
Cost	6728.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -65000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7.8 \cdot 10^{-227}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	19.1%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.4 \cdot 10^{-98} \lor \neg \left(ky \leq 8.2 \cdot 10^{-227}\right):\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \end{array} \]

Alternative 12
Error	19.5%
Cost	584.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -65000000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 9 \cdot 10^{-190}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 13
Error	13.5%
Cost	64.00

\[th \]

Reproduce

herbie shell --seed 2023097 
(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)))