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

Average Error: 0.3 → 0.3

Time: 16.9s

Precision: binary64

Cost: 13760

Math

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

↓

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

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 z)) (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 + z)) * 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 + z)) * 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 + z)) * 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 + z)) * 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 + z)) * 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)
	tmp = ((x * 0.5) - y) * (sqrt((z + z)) * 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 + z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $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

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z + z} \cdot e^{\frac{t \cdot t}{2}}\right)} \]
   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_best_oopsla_all_46_json_45_simplify-74 [=>] 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_best_oopsla_all_46_json_45_simplify-7 [=>] 0.3	\[ \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(e^{\frac{t \cdot t}{2}} \cdot \sqrt{z \cdot 2}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]
   metadata-eval [<=] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot \color{blue}{\left(1 + 1\right)}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   metadata-eval [<=] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot \left(\color{blue}{\frac{2}{2}} + 1\right)} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   metadata-eval [<=] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot \left(\frac{2}{2} + \color{blue}{\frac{2}{2}}\right)} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-23 [<=] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{\color{blue}{\frac{2}{2} \cdot z + z \cdot \frac{2}{2}}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{\color{blue}{z \cdot \frac{2}{2}} + z \cdot \frac{2}{2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   metadata-eval [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot \color{blue}{1} + z \cdot \frac{2}{2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{\color{blue}{z} + z \cdot \frac{2}{2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   metadata-eval [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z + z \cdot \color{blue}{1}} \cdot e^{\frac{t \cdot t}{2}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.3	\[ \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z + \color{blue}{z}} \cdot e^{\frac{t \cdot t}{2}}\right) \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
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 2
Error	1.2
Cost	6976

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

Reproduce

herbie shell --seed 2023090 
(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))))