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

Average Error: 0.3 → 0.3

Time: 12.8s

Precision: binary64

Cost: 20416

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{z + z}\\ \left(\left(x \cdot 0.5\right) \cdot t_1 - t_1 \cdot y\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

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
 (let* ((t_1 (sqrt (+ z z))))
   (* (- (* (* x 0.5) t_1) (* t_1 y)) (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) {
	double t_1 = sqrt((z + z));
	return (((x * 0.5) * t_1) - (t_1 * y)) * 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
    real(8) :: t_1
    t_1 = sqrt((z + z))
    code = (((x * 0.5d0) * t_1) - (t_1 * y)) * 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) {
	double t_1 = Math.sqrt((z + z));
	return (((x * 0.5) * t_1) - (t_1 * y)) * 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):
	t_1 = math.sqrt((z + z))
	return (((x * 0.5) * t_1) - (t_1 * y)) * 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)
	t_1 = sqrt(Float64(z + z))
	return Float64(Float64(Float64(Float64(x * 0.5) * t_1) - Float64(t_1 * y)) * exp(Float64(Float64(t * 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)
	t_1 = sqrt((z + z));
	tmp = (((x * 0.5) * t_1) - (t_1 * y)) * 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_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $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

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

3. Final simplification 0.3

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

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	53.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\left(z + z\right) \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)}\\ \end{array} \]

Alternative 3
Error	52.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 2 \cdot 10^{-241}:\\ \;\;\;\;\left(z + z\right) \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot \left(0.5 \cdot z\right)\right)}\\ \end{array} \]

Alternative 4
Error	1.3
Cost	6976

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

Alternative 5
Error	59.2
Cost	576

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

Alternative 6
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022330 
(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))))