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

Percentage Accurate: 99.3% → 99.8%

Time: 44.4s

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: 99.3% 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 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

4. Applied egg-rr 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: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \sqrt{z} \cdot \mathsf{fma}\left(\sqrt{2}, t\_1 \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right), \left(\sqrt{2} \cdot \left(t\_1 \cdot \mathsf{fma}\left(t, t \cdot 0.020833333333333332, 0.125\right)\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (*
    (sqrt z)
    (fma
     (sqrt 2.0)
     (* t_1 (fma 0.5 (* t t) 1.0))
     (*
      (* (sqrt 2.0) (* t_1 (fma t (* t 0.020833333333333332) 0.125)))
      (* (* t t) (* t t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	return sqrt(z) * fma(sqrt(2.0), (t_1 * fma(0.5, (t * t), 1.0)), ((sqrt(2.0) * (t_1 * fma(t, (t * 0.020833333333333332), 0.125))) * ((t * t) * (t * t))));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	return Float64(sqrt(z) * fma(sqrt(2.0), Float64(t_1 * fma(0.5, Float64(t * t), 1.0)), Float64(Float64(sqrt(2.0) * Float64(t_1 * fma(t, Float64(t * 0.020833333333333332), 0.125))) * Float64(Float64(t * t) * Float64(t * t)))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 95.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 95.5%

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

   1. *-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. *-lowering-*.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-*l* N/A

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

9. Applied egg-rr 95.9%

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

10. Final simplification 95.9%

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

Alternative 3: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\\ \mathsf{fma}\left(t\_1, \sqrt{z}, \sqrt{z} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(t\_1 \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x 0.5) y) (sqrt 2.0))))
   (fma
    t_1
    (sqrt z)
    (*
     (sqrt z)
     (*
      (* t t)
      (* (* t t) (* t_1 (fma (* t t) 0.020833333333333332 0.125))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * 0.5) - y) * sqrt(2.0);
	return fma(t_1, sqrt(z), (sqrt(z) * ((t * t) * ((t * t) * (t_1 * fma((t * t), 0.020833333333333332, 0.125))))));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0))
	return fma(t_1, sqrt(z), Float64(sqrt(z) * Float64(Float64(t * t) * Float64(Float64(t * t) * Float64(t_1 * fma(Float64(t * t), 0.020833333333333332, 0.125))))))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(t$95$1 * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 95.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 95.5%

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

8. Taylor expanded in t around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 95.5

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}, \sqrt{z}, \sqrt{z} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\\ \mathsf{fma}\left(t \cdot t, \sqrt{z} \cdot \left(\left(t \cdot t\right) \cdot \left(t\_1 \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right), \sqrt{z} \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x 0.5) y) (sqrt 2.0))))
   (fma
    (* t t)
    (* (sqrt z) (* (* t t) (* t_1 (fma (* t t) 0.020833333333333332 0.125))))
    (* (sqrt z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * 0.5) - y) * sqrt(2.0);
	return fma((t * t), (sqrt(z) * ((t * t) * (t_1 * fma((t * t), 0.020833333333333332, 0.125)))), (sqrt(z) * t_1));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0))
	return fma(Float64(t * t), Float64(sqrt(z) * Float64(Float64(t * t) * Float64(t_1 * fma(Float64(t * t), 0.020833333333333332, 0.125)))), Float64(sqrt(z) * t_1))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t * t), $MachinePrecision] * N[(N[Sqrt[z], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(t$95$1 * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 95.2%

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

6. Taylor expanded in t around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 95.1

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

8. Simplified 95.1%

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

9. Final simplification 95.1%

   \[\leadsto \mathsf{fma}\left(t \cdot t, \sqrt{z} \cdot \left(\left(t \cdot t\right) \cdot \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right), \sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \]
10. Add Preprocessing

Alternative 5: 94.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
  (fma
   (* t t)
   (fma (* t t) (fma (* t t) 0.020833333333333332 0.125) 0.5)
   1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((2.0 * z))) * fma((t * t), fma((t * t), fma((t * t), 0.020833333333333332, 0.125), 0.5), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z))) * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 0.125), 0.5), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 94.7

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

5. Simplified 94.7%

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

6. Final simplification 94.7%

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

Alternative 6: 92.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
  (fma (* t t) (fma t (* t 0.125) 0.5) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((2.0 * z))) * fma((t * t), fma(t, (t * 0.125), 0.5), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z))) * fma(Float64(t * t), fma(t, Float64(t * 0.125), 0.5), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 94.3

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

5. Simplified 94.3%

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

6. Final simplification 94.3%

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

Alternative 7: 92.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

4. Applied egg-rr 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. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. *-lowering-*.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 93.2

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

7. Simplified 93.2%

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

Alternative 8: 75.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 5e+107)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (sqrt (* 2.0 (* z (fma t t 1.0)))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 5e+107) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((2.0 * (z * fma(t, t, 1.0)))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 5e+107)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 5e+107], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5.0000000000000002e107

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 88.8%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.8

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

   7. Applied egg-rr 88.8%

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

   if 5.0000000000000002e107 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-lowering-exp.f64 N/A

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

      13. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 79.5

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

   7. Simplified 79.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.4

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

   10. Simplified 60.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return fma(0.5, (t * t), 1.0) * (((x * 0.5) - y) * sqrt((2.0 * z)));
}

Julia

function code(x, y, z, t)
	return Float64(fma(0.5, Float64(t * t), 1.0) * Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z))))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 86.5

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

5. Simplified 86.5%

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

6. Final simplification 86.5%

   \[\leadsto \mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \]
7. Add Preprocessing

Alternative 10: 84.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt((2.0 * (z * fma(t, t, 1.0))));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

4. Applied egg-rr 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. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 85.0

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

7. Simplified 85.0%

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

Alternative 11: 57.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

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))
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));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 57.3%

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

   1. *-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 57.3

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

7. Applied egg-rr 57.3%

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

8. Final simplification 57.3%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} \]
9. Add Preprocessing

Alternative 12: 30.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot z} \cdot \left(-y\right) \end{array} \]

FPCore

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

C

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

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 = sqrt((2.0d0 * z)) * -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 57.3%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 30.4

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

8. Simplified 30.4%

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

   1. *-rgt-identity N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \sqrt{z \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{z \cdot 2}} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{z \cdot 2}} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

   6. neg-lowering-neg.f64 30.4

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

10. Applied egg-rr 30.4%

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

11. Final simplification 30.4%

    \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
12. 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 2024198 
(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))))