Toniolo and Linder, Equation (3b), real

Average Error: 4.0 → 2.6

Time: 28.0s

Precision: binary64

Cost: 123208

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\sin ky}{-kx} \cdot \sin th\\ \mathbf{elif}\;t_1 \leq 0.9998:\\ \;\;\;\;t_1 \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

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
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (* (/ (sin ky) (- kx)) (sin th))
     (if (<= t_1 0.9998) (* t_1 (sin th)) (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) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (sin(ky) / -kx) * sin(th);
	} else if (t_1 <= 0.9998) {
		tmp = t_1 * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

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) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sin(ky) / -kx) * Math.sin(th);
	} else if (t_1 <= 0.9998) {
		tmp = t_1 * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

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):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (math.sin(ky) / -kx) * math.sin(th)
	elif t_1 <= 0.9998:
		tmp = t_1 * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

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)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sin(ky) / Float64(-kx)) * sin(th));
	elseif (t_1 <= 0.9998)
		tmp = Float64(t_1 * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
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_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (sin(ky) / -kx) * sin(th);
	elseif (t_1 <= 0.9998)
		tmp = t_1 * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Sin[ky], $MachinePrecision] / (-kx)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9998], N[(t$95$1 * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in kx around 0 64.0

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

   3. Taylor expanded in kx around -inf 46.3

      \[\leadsto \frac{\sin ky}{\color{blue}{-1 \cdot kx}} \cdot \sin th \]

   4. Simplified 46.3

      \[\leadsto \frac{\sin ky}{\color{blue}{-kx}} \cdot \sin th \]
      Proof

      [Start] 46.3	\[ \frac{\sin ky}{-1 \cdot kx} \cdot \sin th \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 46.3	\[ \frac{\sin ky}{\color{blue}{kx \cdot -1}} \cdot \sin th \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 46.3	\[ \frac{\sin ky}{\color{blue}{-kx}} \cdot \sin th \]

   if -inf.0 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) < 0.99980000000000002

   1. Initial program 0.4

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

   if 0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))

   1. Initial program 8.8

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

   2. Taylor expanded in kx around 0 4.4

      \[\leadsto \frac{\sin ky}{\color{blue}{0.5 \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

   3. Taylor expanded in kx around 0 4.4

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -\infty:\\ \;\;\;\;\frac{\sin ky}{-kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9998:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternatives

Alternative 1
Error	18.4
Cost	52040

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	33.2
Cost	45640

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	24.2
Cost	45508

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-22}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \]

Alternative 4
Error	34.7
Cost	32644

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	38.0
Cost	32584

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

Alternative 6
Error	40.5
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;-\frac{\sin ky \cdot th}{kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \end{array} \]

Alternative 7
Error	40.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;-\frac{\sin ky \cdot th}{kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \]

Alternative 8
Error	39.8
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;-\frac{\sin ky \cdot \sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \]

Alternative 9
Error	39.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sin ky}{-kx} \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \]

Alternative 10
Error	42.4
Cost	13252

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

Alternative 11
Error	44.4
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.3 \cdot 10^{-192}:\\ \;\;\;\;\frac{th \cdot ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	51.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -1 \cdot 10^{+44}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{th \cdot ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 13
Error	55.5
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)))