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

Average Error: 0.3 → 0.3

Time: 12.9s

Precision: binary64

Cost: 19968

Math

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

↓

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

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 (* (pow (exp t) t) (* 2.0 z)))))

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

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(((exp(t) ** t) * (2.0d0 * z)))
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((Math.pow(Math.exp(t), t) * (2.0 * z)));
}

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((math.pow(math.exp(t), t) * (2.0 * z)))

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((exp(t) ^ t) * Float64(2.0 * z))))
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(((exp(t) ^ t) * (2.0 * z)));
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[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision] * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

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

3. Applied egg-rr 0.3

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

4. Taylor expanded in z around 0 0.3

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

5. Simplified 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	13632

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

Alternative 2
Error	53.7
Cost	7108

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

Alternative 3
Error	50.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.556859650460364 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(z + z\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 4
Error	1.1
Cost	6976

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

Alternative 5
Error	59.2
Cost	576

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

Alternative 6
Error	60.1
Cost	320

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

Alternative 7
Error	60.1
Cost	192

\[x \cdot z \]

Reproduce

herbie shell --seed 2022228 
(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))))