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

Average Accuracy: 99.5% → 99.5%

Time: 13.8s

Precision: binary64

Cost: 13632

Math

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

↓

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

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 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(((z + 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(((z + 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(((z + 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(((z + 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(Float64(z + 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(((z + 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[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	99.5%
Target	99.5%
Herbie	99.5%

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

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 99.5%

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

   [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* [=>] 99.5	\[ \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]
   associate-*l/ [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{t}{2} \cdot t}}\right) \]
   exp-prod [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{{\left(e^{\frac{t}{2}}\right)}^{t}}\right) \]
   exp-sqrt [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot {\color{blue}{\left(\sqrt{e^{t}}\right)}}^{t}\right) \]

3. Applied egg-rr 99.5%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(0 + \sqrt{\left(z + z\right) \cdot {\left(e^{t}\right)}^{t}}\right)} \]
   Proof

   [Start] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right) \]
   add-log-exp [=>] 7.2	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\log \left(e^{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\right)} \]
   *-un-lft-identity [=>] 7.2	\[ \left(x \cdot 0.5 - y\right) \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\right)} \]
   log-prod [=>] 7.2	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\right)\right)} \]
   metadata-eval [=>] 7.2	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\right)\right) \]
   add-log-exp [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \color{blue}{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\right) \]
   add-sqr-sqrt [=>] 99.0	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \color{blue}{\sqrt{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}} \cdot \sqrt{\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}}}\right) \]
   sqrt-unprod [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \color{blue}{\sqrt{\left(\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right) \cdot \left(\sqrt{z \cdot 2} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}}\right) \]
   swap-sqr [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}}\right) \]
   add-sqr-sqrt [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\color{blue}{\left(z \cdot 2\right)} \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   add-log-exp [=>] 5.0	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\color{blue}{\log \left(e^{z \cdot 2}\right)} \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   exp-lft-sqr [=>] 5.0	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\log \color{blue}{\left(e^{z} \cdot e^{z}\right)} \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   log-prod [=>] 5.0	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\color{blue}{\left(\log \left(e^{z}\right) + \log \left(e^{z}\right)\right)} \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   add-log-exp [<=] 14.0	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\left(\color{blue}{z} + \log \left(e^{z}\right)\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   add-log-exp [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\left(z + \color{blue}{z}\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{t} \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}\right) \]
   pow-prod-down [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\left(z + z\right) \cdot \color{blue}{{\left(\sqrt{e^{t}} \cdot \sqrt{e^{t}}\right)}^{t}}}\right) \]
   add-sqr-sqrt [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\left(z + z\right) \cdot {\color{blue}{\left(e^{t}\right)}}^{t}}\right) \]

4. Simplified 99.5%

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

   [Start] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \left(0 + \sqrt{\left(z + z\right) \cdot {\left(e^{t}\right)}^{t}}\right) \]
   +-lft-identity [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z + z\right) \cdot {\left(e^{t}\right)}^{t}}} \]

5. Taylor expanded in z around 0 99.5%

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

6. Simplified 99.5%

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

   [Start] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \]
   associate-*r* [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]
   unpow2 [=>] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot e^{\color{blue}{t \cdot t}}} \]
   count-2 [<=] 99.5	\[ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z + z\right)} \cdot e^{t \cdot t}} \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	7488

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

Alternative 2
Accuracy	98.7%
Cost	7360

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

Alternative 3
Accuracy	18.8%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 - y \leq 10^{-207}:\\ \;\;\;\;z \cdot \left(x + y \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	33.5%
Cost	7176

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-t_1\\ \end{array} \]

Alternative 5
Accuracy	23.3%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 10^{-120}:\\ \;\;\;\;\sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	98.2%
Cost	6976

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

Alternative 7
Accuracy	7.4%
Cost	448

\[z \cdot \left(x + y \cdot -2\right) \]

Alternative 8
Accuracy	5.9%
Cost	320

\[\left(y \cdot z\right) \cdot -2 \]

Alternative 9
Accuracy	6.0%
Cost	192

\[x \cdot z \]

Reproduce

herbie shell --seed 2023136 
(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))))