Toniolo and Linder, Equation (3b), real

Average Error: 3.9 → 0.2

Time: 41.0s

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 3.9

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

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)): 20 points increase in error, 2 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	19.0
Cost	58580

\[\begin{array}{l} t_1 := \left|\sin ky\right|\\ t_2 := \frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{if}\;\sin th \leq -0.6948:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq -0.005:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 10^{-24}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\sin th \leq 0.35:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 0.64:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	18.9
Cost	58580

\[\begin{array}{l} t_1 := \left|\sin ky\right|\\ t_2 := \frac{\sin ky \cdot \sin th}{\left|\sin kx\right|}\\ \mathbf{if}\;\sin th \leq -0.6948:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq -0.005:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 10^{-24}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin th \leq 0.35:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 0.64:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	39.4
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-10}:\\ \;\;\;\;ky \cdot \frac{th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-99}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 4
Error	39.4
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-10}:\\ \;\;\;\;ky \cdot \frac{th}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-121}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-99}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 5
Error	38.3
Cost	39116

\[\begin{array}{l} t_1 := \sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-201}:\\ \;\;\;\;th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	25.7
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{th}{\frac{\left|\sin ky\right|}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	20.7
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sin ky}{\frac{\left|\sin ky\right|}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	20.7
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin ky\right|}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	0.3
Cost	32384

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

Alternative 10
Error	30.2
Cost	19784

\[\begin{array}{l} \mathbf{if}\;ky \leq -1700000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.42 \cdot 10^{-53}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	41.1
Cost	13648

\[\begin{array}{l} t_1 := ky \cdot \frac{th}{\left|\sin kx\right|}\\ \mathbf{if}\;ky \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.3 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.4 \cdot 10^{-219}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 5.5 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	41.1
Cost	13648

\[\begin{array}{l} t_1 := th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{if}\;ky \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.65 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 4.4 \cdot 10^{-215}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 9.6 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	42.6
Cost	7116

\[\begin{array}{l} \mathbf{if}\;ky \leq -9 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -9.8 \cdot 10^{-104}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 2.25 \cdot 10^{-207}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	42.6
Cost	7116

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -9 \cdot 10^{-104}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;ky \leq 2.2 \cdot 10^{-207}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	42.6
Cost	7116

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;ky \leq 2.05 \cdot 10^{-207}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	42.6
Cost	7116

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -4.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;ky \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	42.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Error	44.2
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19
Error	50.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -8.2 \cdot 10^{-30}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 20
Error	50.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.85 \cdot 10^{-30}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.4 \cdot 10^{-127}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 21
Error	55.1
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022325 
(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)))