Ian Simplification

Average Error: 59.6 → 59.6

Time: 22.8s

Precision: binary64

Cost: 139648

Math

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ t_1 := \frac{\pi}{2} + -2 \cdot t_0\\ t_2 := \frac{-1}{t_1} \cdot \left(\left(2 \cdot t_0 - \frac{\pi}{2}\right) \cdot t_1\right)\\ t_2 \cdot \left(t_2 \cdot \frac{1}{\frac{\pi}{2} + t_0 \cdot -2}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (/ (- 1.0 x) 2.0))))
        (t_1 (+ (/ PI 2.0) (* -2.0 t_0)))
        (t_2 (* (/ -1.0 t_1) (* (- (* 2.0 t_0) (/ PI 2.0)) t_1))))
   (* t_2 (* t_2 (/ 1.0 (+ (/ PI 2.0) (* t_0 -2.0)))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

↓

double code(double x) {
	double t_0 = asin(sqrt(((1.0 - x) / 2.0)));
	double t_1 = (((double) M_PI) / 2.0) + (-2.0 * t_0);
	double t_2 = (-1.0 / t_1) * (((2.0 * t_0) - (((double) M_PI) / 2.0)) * t_1);
	return t_2 * (t_2 * (1.0 / ((((double) M_PI) / 2.0) + (t_0 * -2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

↓

public static double code(double x) {
	double t_0 = Math.asin(Math.sqrt(((1.0 - x) / 2.0)));
	double t_1 = (Math.PI / 2.0) + (-2.0 * t_0);
	double t_2 = (-1.0 / t_1) * (((2.0 * t_0) - (Math.PI / 2.0)) * t_1);
	return t_2 * (t_2 * (1.0 / ((Math.PI / 2.0) + (t_0 * -2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

↓

def code(x):
	t_0 = math.asin(math.sqrt(((1.0 - x) / 2.0)))
	t_1 = (math.pi / 2.0) + (-2.0 * t_0)
	t_2 = (-1.0 / t_1) * (((2.0 * t_0) - (math.pi / 2.0)) * t_1)
	return t_2 * (t_2 * (1.0 / ((math.pi / 2.0) + (t_0 * -2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

↓

function code(x)
	t_0 = asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))
	t_1 = Float64(Float64(pi / 2.0) + Float64(-2.0 * t_0))
	t_2 = Float64(Float64(-1.0 / t_1) * Float64(Float64(Float64(2.0 * t_0) - Float64(pi / 2.0)) * t_1))
	return Float64(t_2 * Float64(t_2 * Float64(1.0 / Float64(Float64(pi / 2.0) + Float64(t_0 * -2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

↓

function tmp = code(x)
	t_0 = asin(sqrt(((1.0 - x) / 2.0)));
	t_1 = (pi / 2.0) + (-2.0 * t_0);
	t_2 = (-1.0 / t_1) * (((2.0 * t_0) - (pi / 2.0)) * t_1);
	tmp = t_2 * (t_2 * (1.0 / ((pi / 2.0) + (t_0 * -2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi / 2.0), $MachinePrecision] + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 / t$95$1), $MachinePrecision] * N[(N[(N[(2.0 * t$95$0), $MachinePrecision] - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(t$95$2 * N[(t$95$2 * N[(1.0 / N[(N[(Pi / 2.0), $MachinePrecision] + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	59.6
Target	0
Herbie	59.6

\[\sin^{-1} x \]

Derivation

1. Initial program 59.6

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Applied egg-rr 59.6

   \[\leadsto \color{blue}{\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right)} \]

3. Applied egg-rr 59.6

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right)\right)} \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]

4. Simplified 59.6

   \[\leadsto \color{blue}{\left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right)} \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   Proof

   [Start] 59.6	\[ \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right)\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \color{blue}{\left(\frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2} \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)}\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 59.6	\[ \color{blue}{\left(\frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)\right)} \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]

5. Applied egg-rr 59.6

   \[\leadsto \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right)\right)} \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]

6. Simplified 59.6

   \[\leadsto \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right)} \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   Proof

   [Start] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right) \cdot \frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right)\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \color{blue}{\left(\frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2} \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)}\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)\right)} \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{\frac{\pi}{2} + \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\right)\right)\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.6	\[ \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + \color{blue}{-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)\right)\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]

7. Final simplification 59.6

   \[\leadsto \left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \left(\left(\frac{-1}{\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \left(\left(2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \frac{\pi}{2}\right) \cdot \left(\frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right)\right) \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2}\right) \]

Alternatives

Alternative 1
Error	59.6
Cost	99712

\[\begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ t_1 := \frac{\pi}{2} + -2 \cdot t_0\\ t_2 := \frac{\pi}{2} + t_0 \cdot -2\\ \left(\frac{-1}{t_1} \cdot \left(\left(2 \cdot t_0 - \frac{\pi}{2}\right) \cdot t_1\right)\right) \cdot \left(t_2 \cdot \frac{1}{t_2}\right) \end{array} \]

Alternative 2
Error	59.6
Cost	99712

\[\begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ t_1 := \frac{\pi}{2} + t_0 \cdot -2\\ t_2 := \frac{\pi}{2} + -2 \cdot t_0\\ t_1 \cdot \left(\left(\frac{-1}{t_2} \cdot \left(\left(2 \cdot t_0 - \frac{\pi}{2}\right) \cdot t_2\right)\right) \cdot \frac{1}{t_1}\right) \end{array} \]

Alternative 3
Error	59.6
Cost	59776

\[\begin{array}{l} t_0 := \frac{\pi}{2} + \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot -2\\ t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right) \end{array} \]

Alternative 4
Error	59.6
Cost	19840

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

Alternative 5
Error	61.4
Cost	19584

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))