Toniolo and Linder, Equation (3b), real

Average Error: 3.8 → 0.2

Time: 46.8s

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 3.8

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]
   Proof
   (*.f64 (/.f64 (sin.f64 ky) (hypot.f64 (sin.f64 ky) (sin.f64 kx))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 ky) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (sin.f64 ky) (sin.f64 ky)) (*.f64 (sin.f64 kx) (sin.f64 kx)))))) (sin.f64 th)): 19 points increase in error, 4 points decrease in error
   (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 ky) 2)) (*.f64 (sin.f64 kx) (sin.f64 kx))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 ky) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 kx) 2))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	15.0
Cost	52112

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1} \cdot th\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.975:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.999:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	15.0
Cost	52112

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.975:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.999:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	15.0
Cost	52112

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \frac{1}{t_1}\right)\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.975:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.999:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	14.9
Cost	52112

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.975:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.999:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	25.1
Cost	45776

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.14:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{elif}\;\sin th \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th}}\\ \mathbf{elif}\;\sin th \leq 0.775:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	25.1
Cost	45776

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.14:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{elif}\;\sin th \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin th \leq 0.775:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.985:\\ \;\;\;\;\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	34.4
Cost	32716

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-172}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	0.3
Cost	32384

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

Alternative 9
Error	35.5
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	38.9
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.3 \cdot 10^{-17}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	43.8
Cost	7248

\[\begin{array}{l} t_1 := th \cdot \frac{ky}{\sin kx}\\ \mathbf{if}\;ky \leq -1020000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.1 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{-209}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	43.8
Cost	7248

\[\begin{array}{l} \mathbf{if}\;ky \leq -1020000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.32 \cdot 10^{-191}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-210}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	44.0
Cost	7248

\[\begin{array}{l} t_1 := \frac{ky \cdot th}{\sin kx}\\ \mathbf{if}\;ky \leq -1020000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -5.3 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-208}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	45.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4 \cdot 10^{-119}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	42.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.45 \cdot 10^{-119}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	49.1
Cost	6464

\[\sin th \]

Alternative 17
Error	55.4
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022318 
(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)))