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

Percentage Accurate: 99.4% → 99.8%

Time: 20.2s

Precision: binary64

Cost: 20032

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt(exp((t * t))) * sqrt((z * 2.0d0))) * ((x * 0.5d0) - y)
end function

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 (Math.sqrt(Math.exp((t * t))) * Math.sqrt((z * 2.0))) * ((x * 0.5) - y);
}

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

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(sqrt(exp(Float64(t * t))) * sqrt(Float64(z * 2.0))) * Float64(Float64(x * 0.5) - y))
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 = (sqrt(exp((t * t))) * sqrt((z * 2.0))) * ((x * 0.5) - y);
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[Sqrt[N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Target

Original	99.4%
Target	99.4%
Herbie	99.8%

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

Derivation

1. Initial program 98.7%

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

2. Simplified 99.8%

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

   [Start] 98.7%	\[ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   associate-*l* [=>] 99.8%	\[ \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]
   exp-sqrt [=>] 99.8%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	20032

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

Alternative 2
Accuracy	92.4%
Cost	14024

\[\begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 10^{-10}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z + t \cdot \left(z \cdot t\right)\right)}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\sqrt{e^{t \cdot t} \cdot \left(z \cdot 2\right)}}{\frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	13632

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

Alternative 4
Accuracy	79.3%
Cost	7360

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

Alternative 5
Accuracy	83.5%
Cost	7360

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

Alternative 6
Accuracy	31.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq -520000000:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{z + z}}{\frac{-1}{y}}\\ \end{array} \]

Alternative 7
Accuracy	55.7%
Cost	6976

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

Alternative 8
Accuracy	16.4%
Cost	6848

\[\sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)} \]

Reproduce

herbie shell --seed 2023272 
(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))))