Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Average Accuracy: 99.6% → 99.6%

Time: 16.8s

Precision: binary64

Cost: 20160

Math

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

↓

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {e}^{\left(0.5 \cdot \left(t \cdot t\right)\right)} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

↓

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow E (* 0.5 (* t t)))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

↓

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(((double) M_E), (0.5 * (t * t)));
}

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

↓

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.E, (0.5 * (t * t)));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

↓

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.e, (0.5 * (t * t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1) ^ Float64(0.5 * Float64(t * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

↓

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (2.71828182845904523536 ^ (0.5 * (t * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[E, N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	99.6%
Target	99.6%
Herbie	99.6%

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \]

Derivation

1. Initial program 99.6%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Applied egg-rr 99.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(0.5 \cdot \left(t \cdot t\right)\right)}} \]

3. Final simplification 99.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {e}^{\left(0.5 \cdot \left(t \cdot t\right)\right)} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	20096

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{t}\right)}^{\left(0.5 \cdot t\right)} \]

Alternative 2
Accuracy	99.6%
Cost	13760

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

Alternative 3
Accuracy	99.6%
Cost	13632

\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}} \]

Alternative 4
Accuracy	98.5%
Cost	7488

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(1 + t \cdot \left(0.5 \cdot t\right)\right) \]

Alternative 5
Accuracy	98.5%
Cost	7488

\[\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(1 + t \cdot \left(0.5 \cdot t\right)\right)\right) \]

Alternative 6
Accuracy	98.5%
Cost	7360

\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)} \]

Alternative 7
Accuracy	54.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-162} \lor \neg \left(y \leq 2.8 \cdot 10^{-211}\right):\\ \;\;\;\;\frac{\sqrt{z \cdot 2}}{\frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot \left(0.5 \cdot z\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	72.9%
Cost	7113

\[\begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-31} \lor \neg \left(x \leq 2.05 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t_1}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{-1}{y}}\\ \end{array} \]

Alternative 9
Accuracy	98.0%
Cost	6976

\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z} \]

Alternative 10
Accuracy	16.7%
Cost	6848

\[\sqrt{x \cdot \left(x \cdot \left(0.5 \cdot z\right)\right)} \]

Alternative 11
Accuracy	3.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023141 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))