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

Percentage Accurate: 99.5% → 99.8%

Time: 24.3s

Precision: binary64

Cost: 13632

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 (* 2.0 (* z (exp (* 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((2.0 * (z * exp((t * t)))));
}

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 = ((x * 0.5d0) - y) * sqrt((2.0d0 * (z * exp((t * t)))))
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 ((x * 0.5) - y) * Math.sqrt((2.0 * (z * Math.exp((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((2.0 * (z * math.exp((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(x * 0.5) - y) * sqrt(Float64(2.0 * Float64(z * exp(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((2.0 * (z * exp((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[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	99.5%
Target	99.5%
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 99.5%

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

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] 99.5%	\[ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. Applied egg-rr 75.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)} - 1\right)} \]
   Step-by-step derivation

   [Start] 99.8%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right) \]
   expm1-log1p-u [=>] 98.6%	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]
   expm1-udef [=>] 75.4%	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]
   sqrt-unprod [=>] 75.4%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]
   associate-*l* [=>] 75.4%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]
   exp-prod [=>] 75.4%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)}\right)} - 1\right) \]

4. Simplified 99.9%

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

   [Start] 75.4%	\[ \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)} - 1\right) \]
   expm1-def [=>] 98.6%	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)\right)} \]
   expm1-log1p [=>] 99.9%	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \]
   *-commutative [=>] 99.9%	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot {\left(e^{t}\right)}^{t}\right) \cdot z}} \]
   associate-*l* [=>] 99.9%	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot \left({\left(e^{t}\right)}^{t} \cdot z\right)}} \]
   *-commutative [<=] 99.9%	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \color{blue}{\left(z \cdot {\left(e^{t}\right)}^{t}\right)}} \]
   exp-prod [<=] 99.9%	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \color{blue}{e^{t \cdot t}}\right)} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13632

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

Alternative 2
Accuracy	90.3%
Cost	14152

\[\begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := 0.5 \cdot \left(t \cdot t\right)\\ t_3 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t_3 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+137}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot y\right)} \cdot e^{\frac{t \cdot t}{2}}\\ \mathbf{elif}\;t \cdot t \leq 10^{+264}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot \frac{1 - t_2 \cdot t_2}{\left(t \cdot t\right) \cdot -0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot t_2\right)\\ \end{array} \]

Alternative 3
Accuracy	90.0%
Cost	8776

\[\begin{array}{l} t_1 := 0.5 \cdot \left(t \cdot t\right)\\ t_2 := x \cdot 0.5 - y\\ t_3 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+117}:\\ \;\;\;\;t_2 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+264}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \frac{1 - t_1 \cdot t_1}{\left(t \cdot t\right) \cdot -0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot t_1\right)\\ \end{array} \]

Alternative 4
Accuracy	90.2%
Cost	8644

\[\begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := 0.5 \cdot \left(t \cdot t\right)\\ t_3 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 10^{+264}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot \frac{1 - t_2 \cdot t_2}{1 - t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot t_2\right)\\ \end{array} \]

Alternative 5
Accuracy	86.4%
Cost	7620

\[\begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	86.1%
Cost	7488

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

Alternative 7
Accuracy	86.7%
Cost	7488

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

Alternative 8
Accuracy	84.9%
Cost	7360

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

Alternative 9
Accuracy	57.2%
Cost	6976

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

Reproduce

herbie shell --seed 2023263 
(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))))