Toniolo and Linder, Equation (3b), real

Average Error: 6.19% → 0.32%

Time: 31.1s

Precision: binary64

Cost: 32384

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 6.19

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

2. Simplified 0.32

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

   [Start] 6.19	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 6.19	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 6.19	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 6.19	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 0.32	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Final simplification 0.32

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

Alternatives

Alternative 1
Error	55.73%
Cost	45776

\[\begin{array}{l} t_1 := \frac{\sin th}{\sin kx} \cdot \left|\sin ky\right|\\ \mathbf{if}\;\sin kx \leq -0.586:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -0.125:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \end{array} \]

Alternative 2
Error	56.58%
Cost	32712

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \end{array} \]

Alternative 3
Error	55.69%
Cost	32712

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \end{array} \]

Alternative 4
Error	56.36%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 5
Error	56.36%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 6
Error	26.61%
Cost	26633

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -7500000000000 \lor \neg \left(th \leq 2.1 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\frac{\sin th}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \]

Alternative 7
Error	62.88%
Cost	26312

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\frac{1}{ky}}}{\sin kx}\\ \end{array} \]

Alternative 8
Error	26.86%
Cost	26249

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -7500000000000 \lor \neg \left(th \leq 6.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \]

Alternative 9
Error	39.82%
Cost	26248

\[\begin{array}{l} \mathbf{if}\;th \leq -0.126:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;th \leq 1.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \left|\sin ky\right|\\ \end{array} \]

Alternative 10
Error	62.5%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 11
Error	62.49%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 12
Error	53.98%
Cost	19916

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ \mathbf{if}\;ky \leq -13800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -9.6 \cdot 10^{-305}:\\ \;\;\;\;\sin th \cdot t_1\\ \mathbf{elif}\;ky \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\sin th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	66.37%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.3 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	66.27%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	66.21%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-179}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	68.43%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.08 \cdot 10^{-14}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.9 \cdot 10^{-192}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	78.92%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.72 \cdot 10^{+41}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-192}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 18
Error	78.92%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 9.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 19
Error	86.68%
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023090 
(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)))