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

Percentage Accurate: 99.4% → 99.8%

Time: 19.3s

Alternatives: 16

Speedup: 1.1×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

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));
}

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

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));
}

Python

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

MATLAB

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]

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

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));
}

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

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));
}

Python

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

MATLAB

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]

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}} \cdot \mathsf{fma}\left(0.5, x, -y\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (sqrt (* (* 2.0 z) (exp (* t t)))) (fma 0.5 x (- y))))

C

double code(double x, double y, double z, double t) {
	return sqrt(((2.0 * z) * exp((t * t)))) * fma(0.5, x, -y);
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(Float64(2.0 * z) * exp(Float64(t * t)))) * fma(0.5, x, Float64(-y)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(y\right)\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\color{blue}{\sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]

   12. lift-exp.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{\frac{t \cdot t}{2}}}\right) \]

   13. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{t \cdot t}{2}}}\right) \]

   14. exp-sqrt N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   15. sqrt-unprod N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

   16. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

   19. lower-exp.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]
6. Add Preprocessing

Alternative 2: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{-16}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (fma 0.5 x (- y))))
   (if (<= (* t t) 2e-16)
     (* t_1 t_2)
     (if (<= (* t t) 5e+77)
       (* (* x 0.5) (sqrt (* (* 2.0 z) (exp (* t t)))))
       (*
        (*
         t_1
         (fma
          (fma (fma (* t t) 0.020833333333333332 0.125) (* t t) 0.5)
          (* t t)
          1.0))
        t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = fma(0.5, x, -y);
	double tmp;
	if ((t * t) <= 2e-16) {
		tmp = t_1 * t_2;
	} else if ((t * t) <= 5e+77) {
		tmp = (x * 0.5) * sqrt(((2.0 * z) * exp((t * t))));
	} else {
		tmp = (t_1 * fma(fma(fma((t * t), 0.020833333333333332, 0.125), (t * t), 0.5), (t * t), 1.0)) * t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = fma(0.5, x, Float64(-y))
	tmp = 0.0
	if (Float64(t * t) <= 2e-16)
		tmp = Float64(t_1 * t_2);
	elseif (Float64(t * t) <= 5e+77)
		tmp = Float64(Float64(x * 0.5) * sqrt(Float64(Float64(2.0 * z) * exp(Float64(t * t)))));
	else
		tmp = Float64(Float64(t_1 * fma(fma(fma(Float64(t * t), 0.020833333333333332, 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0)) * t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * x + (-y)), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e-16], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+77], N[(N[(x * 0.5), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 2e-16

   1. Initial program 99.6%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(y\right)\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\color{blue}{\sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{\frac{t \cdot t}{2}}}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{t \cdot t}{2}}}\right) \]

      14. exp-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

      19. lower-exp.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

      2. lower-*.f64 99.6

         \[\leadsto \mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   if 2e-16 < (*.f64 t t) < 5.00000000000000004e77

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(y\right)\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\color{blue}{\sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{\frac{t \cdot t}{2}}}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{t \cdot t}{2}}}\right) \]

      14. exp-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

      15. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}} \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \sqrt{\color{blue}{e^{t \cdot t} \cdot \left(z \cdot 2\right)}} \]

      19. lower-exp.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \sqrt{e^{t \cdot t} \cdot \left(z \cdot 2\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \sqrt{e^{t \cdot t} \cdot \left(z \cdot 2\right)} \]

      2. lower-*.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if 5.00000000000000004e77 < (*.f64 t t)

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, {t}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2}} + \frac{1}{2}, {t}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right)}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

      14. lower-*.f64 97.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

   5. Applied rewrites 97.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(y\right)\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right)} \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right) \cdot \sqrt{z \cdot 2}\right)} \]

      12. lower-*.f64 98.7

          \[\leadsto \mathsf{fma}\left(0.5, x, -y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \sqrt{z \cdot 2}\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, -y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \sqrt{2 \cdot z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((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(1.0d0) ** ((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.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((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(1.0) ^ 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(1.0) ^ ((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[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))