Toniolo and Linder, Equation (3b), real

Average Error: 4.1 → 0.2

Time: 45.7s

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 4.1

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

2. Taylor expanded in kx around inf 4.1

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

3. Simplified 0.2

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   Proof
   (hypot.f64 (sin.f64 ky) (sin.f64 kx)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (sin.f64 ky) (sin.f64 ky)) (*.f64 (sin.f64 kx) (sin.f64 kx))))): 7 points increase in error, 5 points decrease in error
   (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 ky) 2)) (*.f64 (sin.f64 kx) (sin.f64 kx)))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (pow.f64 (sin.f64 ky) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 kx) 2)))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	14.6
Cost	58644

\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;\sin kx \leq -0.88:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{0.5 + \cos \left(kx \cdot 2\right) \cdot -0.5}}\right)\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 2
Error	14.5
Cost	58644

\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;\sin kx \leq -0.88:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 3
Error	14.5
Cost	58644

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin kx \leq -0.88:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 4
Error	14.5
Cost	58644

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin kx \leq -0.88:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 5
Error	14.6
Cost	58644

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin kx \leq -0.88:\\ \;\;\;\;\frac{1}{\frac{t_1}{\sin ky \cdot th}}\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 6
Error	13.2
Cost	52112

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 + \cos \left(kx \cdot 2\right) \cdot -0.5}}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 0.87:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 0.964:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 7
Error	34.6
Cost	46160

\[\begin{array}{l} t_1 := \sin th \cdot \left(ky \cdot \sqrt{\frac{1}{0.5 + \cos \left(kx \cdot 2\right) \cdot -0.5}}\right)\\ \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	14.3
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \sqrt{\frac{1}{0.5 + \cos \left(kx \cdot 2\right) \cdot -0.5}}\right)\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]

Alternative 9
Error	0.2
Cost	32384

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

Alternative 10
Error	0.3
Cost	32384

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

Alternative 11
Error	34.9
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	41.6
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	38.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.4198306224642115 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.826244104623282 \cdot 10^{-126}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	43.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.48310294439298 \cdot 10^{-12}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4554306753922401 \cdot 10^{-168}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	45.7
Cost	6920

\[\begin{array}{l} \mathbf{if}\;ky \leq -5.436975003526325 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4554306753922401 \cdot 10^{-168}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	48.8
Cost	6464

\[\sin th \]

Alternative 17
Error	55.4
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022315 
(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)))