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

Average Error: 0.3 → 0.3

Time: 31.7s

Precision: binary64

Cost: 13760

Math

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

↓

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

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 2.0)) (exp (* t (/ t 2.0))))))

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

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 * 2.0d0)) * exp((t * (t / 2.0d0))))
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 * 2.0)) * Math.exp((t * (t / 2.0))));
}

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 * 2.0)) * math.exp((t * (t / 2.0))))

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) * Float64(sqrt(Float64(z * 2.0)) * exp(Float64(t * Float64(t / 2.0)))))
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 * 2.0)) * exp((t * (t / 2.0))));
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[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t * N[(t / 2.0), $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. Simplified 0.3

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

   [Start] 0.3	\[ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   rational.json-simplify-2 [=>] 0.3	\[ \color{blue}{e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)} \]
   rational.json-simplify-43 [=>] 0.3	\[ \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]
   rational.json-simplify-49 [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{t \cdot \frac{t}{2}}}\right) \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.1
Cost	13956

\[\begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := 0.5 \cdot \left(x \cdot t_1\right)\\ \mathbf{if}\;t \cdot t \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\left(-y\right) \cdot t_1 + t_2\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	13892

\[\begin{array}{l} \mathbf{if}\;t \cdot t \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(x + \left(x - y \cdot 4\right)\right) \cdot \sqrt{z + z}}{4}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right) \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	13760

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

Alternative 4
Error	19.0
Cost	7640

\[\begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := 0.5 \cdot \left(x \cdot t_1\right)\\ t_3 := \left(-y\right) \cdot t_1\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+96}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-94}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+94}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	1.3
Cost	7232

\[\frac{\left(x + \left(x - y \cdot 4\right)\right) \cdot \sqrt{z + z}}{4} \]

Alternative 6
Error	1.2
Cost	6976

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

Alternative 7
Error	31.8
Cost	6784

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

Reproduce

herbie shell --seed 2023074 
(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))))