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

Percentage Accurate: 99.4% → 99.8%

Time: 36.3s

Alternatives: 12

Speedup: 1.1×

• 44.5% 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

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt((2.0 * (z * exp((t * 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((2.0d0 * (z * exp((t * t)))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * Math.sqrt((2.0 * (z * Math.exp((t * t)))));
}

Python

def code(x, y, z, t):
	return ((x * 0.5) - y) * math.sqrt((2.0 * (z * math.exp((t * t)))))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * sqrt((2.0 * (z * exp((t * t)))));
end

Wolfram

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

Derivation

1. Initial program 98.6%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. exp-sqrt N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 88.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, 0.5 \cdot t, 1\right)\\ t_2 := \sqrt{2 \cdot z}\\ t_3 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+139}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot t\_1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+254}:\\ \;\;\;\;\left(y \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right) \cdot \left(-t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma t (* 0.5 t) 1.0))
        (t_2 (sqrt (* 2.0 z)))
        (t_3 (- (* x 0.5) y)))
   (if (<= (* t t) 5e+139)
     (* t_2 (* t_3 t_1))
     (if (<= (* t t) 1e+254)
       (* (* y (fma (* t t) (fma t (* t 0.125) 0.5) 1.0)) (- t_2))
       (* t_3 (* t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, (0.5 * t), 1.0);
	double t_2 = sqrt((2.0 * z));
	double t_3 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 5e+139) {
		tmp = t_2 * (t_3 * t_1);
	} else if ((t * t) <= 1e+254) {
		tmp = (y * fma((t * t), fma(t, (t * 0.125), 0.5), 1.0)) * -t_2;
	} else {
		tmp = t_3 * (t_2 * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, Float64(0.5 * t), 1.0)
	t_2 = sqrt(Float64(2.0 * z))
	t_3 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 5e+139)
		tmp = Float64(t_2 * Float64(t_3 * t_1));
	elseif (Float64(t * t) <= 1e+254)
		tmp = Float64(Float64(y * fma(Float64(t * t), fma(t, Float64(t * 0.125), 0.5), 1.0)) * Float64(-t_2));
	else
		tmp = Float64(t_3 * Float64(t_2 * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(0.5 * t), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e+139], N[(t$95$2 * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+254], N[(N[(y * N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * (-t$95$2)), $MachinePrecision], N[(t$95$3 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 5.0000000000000003e139

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 90.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.9%

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

   if 5.0000000000000003e139 < (*.f64 t t) < 9.9999999999999994e253

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.9%

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

   10. Applied rewrites 73.9%

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

   if 9.9999999999999994e253 < (*.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 96.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 96.6

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

   12. Applied rewrites 96.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t, 0.5 \cdot t, 1\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+254}:\\ \;\;\;\;\left(y \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right) \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(t, 0.5 \cdot t, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.4% 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 2024228 
(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))))