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

Average Error: 0.3 → 0.3

Time: 16.0s

Precision: binary64

Cost: 20224

Math

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

↓

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

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 (* (+ z z) (pow (exp (+ t t)) (* 0.5 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) * pow(exp((t + t)), (0.5 * 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)) ** (0.5d0 * 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.pow(Math.exp((t + t)), (0.5 * 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.pow(math.exp((t + t)), (0.5 * 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)) ^ Float64(0.5 * 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)) ^ (0.5 * 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[Power[N[Exp[N[(t + t), $MachinePrecision]], $MachinePrecision], N[(0.5 * t), $MachinePrecision]], $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. Applied egg-rr 0.3

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

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	19968

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

Alternative 2
Error	0.3
Cost	13632

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

Alternative 3
Error	0.8
Cost	7488

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

Alternative 4
Error	17.3
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := y \cdot \left(-t_1\right)\\ \mathbf{if}\;y \leq -126593.76765721013:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.590912172145655 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	1.1
Cost	6976

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

Alternative 6
Error	31.9
Cost	6784

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

Alternative 7
Error	61.9
Cost	6720

\[y \cdot \sqrt{z \cdot 2} \]

Reproduce

herbie shell --seed 2022300 
(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))))