?

Average Error: 59.7 → 58.8
Time: 26.5s
Precision: binary64
Cost: 20096

?

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
↓
\[\left(\pi \cdot 0.5 + 1\right) + \left(-1 - 2 \cdot \sin^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right) \]
(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))
↓
(FPCore (x)
 :precision binary64
 (+ (+ (* PI 0.5) 1.0) (- -1.0 (* 2.0 (asin (sqrt (* (- 1.0 x) 0.5)))))))
double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}
↓
double code(double x) {
	return ((((double) M_PI) * 0.5) + 1.0) + (-1.0 - (2.0 * asin(sqrt(((1.0 - x) * 0.5)))));
}

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) {
	return ((Math.PI * 0.5) + 1.0) + (-1.0 - (2.0 * Math.asin(Math.sqrt(((1.0 - x) * 0.5)))));
}

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))
↓
def code(x):
	return ((math.pi * 0.5) + 1.0) + (-1.0 - (2.0 * math.asin(math.sqrt(((1.0 - x) * 0.5)))))

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)
	return Float64(Float64(Float64(pi * 0.5) + 1.0) + Float64(-1.0 - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) * 0.5))))))
end

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end
↓
function tmp = code(x)
	tmp = ((pi * 0.5) + 1.0) + (-1.0 - (2.0 * asin(sqrt(((1.0 - x) * 0.5)))));
end

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_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + N[(-1.0 - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)
↓
\left(\pi \cdot 0.5 + 1\right) + \left(-1 - 2 \cdot \sin^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.7
Target0.0
Herbie58.8
\[\sin^{-1} x \]

Derivation?

  1. Initial program 59.7

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

  2. Applied egg-rr58.8

    \[\leadsto \color{blue}{\left(\pi \cdot 0.5 + 1\right) + \left(-1 - 2 \cdot \sin^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\right)} \]

  3. Final simplification58.8

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

Alternatives

Alternative 1
Error59.7
Cost19840
\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

Error

Reproduce?

herbie shell --seed 2023073 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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