Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 8

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + 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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + 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 = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (fma (* z -0.5) y (* 0.125 x)) t))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-neg-in N/A

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

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

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

   7. div-inv N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -0.5, y, 0.125 \cdot x\right)} + t \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, 0.125 \cdot x\right)\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z y) -5e+123)
   (fma y (* z -0.5) (* 0.125 x))
   (if (<= (* z y) 5e+30) (fma 0.125 x t) (fma y (* z -0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * y) <= -5e+123) {
		tmp = fma(y, (z * -0.5), (0.125 * x));
	} else if ((z * y) <= 5e+30) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = fma(y, (z * -0.5), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * y) <= -5e+123)
		tmp = fma(y, Float64(z * -0.5), Float64(0.125 * x));
	elseif (Float64(z * y) <= 5e+30)
		tmp = fma(0.125, x, t);
	else
		tmp = fma(y, Float64(z * -0.5), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * y), $MachinePrecision], -5e+123], N[(y * N[(z * -0.5), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 5e+30], N[(0.125 * x + t), $MachinePrecision], N[(y * N[(z * -0.5), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -4.99999999999999974e123

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 90.8

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

   5. Simplified 90.8%

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

   if -4.99999999999999974e123 < (*.f64 y z) < 4.9999999999999998e30

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

      2. accelerator-lowering-fma.f64 91.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]

   if 4.9999999999999998e30 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, 0.125 \cdot x\right)\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (* z -0.5) t)))
   (if (<= (* z y) -5e+101) t_1 (if (<= (* z y) 5e+30) (fma 0.125 x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z * -0.5), t);
	double tmp;
	if ((z * y) <= -5e+101) {
		tmp = t_1;
	} else if ((z * y) <= 5e+30) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z * -0.5), t)
	tmp = 0.0
	if (Float64(z * y) <= -5e+101)
		tmp = t_1;
	elseif (Float64(z * y) <= 5e+30)
		tmp = fma(0.125, x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -5e+101], t$95$1, If[LessEqual[N[(z * y), $MachinePrecision], 5e+30], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -4.99999999999999989e101 or 4.9999999999999998e30 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 88.2

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

   5. Simplified 88.2%

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

   if -4.99999999999999989e101 < (*.f64 y z) < 4.9999999999999998e30

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

      2. accelerator-lowering-fma.f64 92.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot -0.5\right) \cdot y\\ \mathbf{if}\;z \cdot y \leq -4 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot y \leq 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z -0.5) y)))
   (if (<= (* z y) -4e+181) t_1 (if (<= (* z y) 1e+172) (fma 0.125 x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) * y;
	double tmp;
	if ((z * y) <= -4e+181) {
		tmp = t_1;
	} else if ((z * y) <= 1e+172) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) * y)
	tmp = 0.0
	if (Float64(z * y) <= -4e+181)
		tmp = t_1;
	elseif (Float64(z * y) <= 1e+172)
		tmp = fma(0.125, x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -4e+181], t$95$1, If[LessEqual[N[(z * y), $MachinePrecision], 1e+172], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -3.9999999999999997e181 or 1.0000000000000001e172 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 92.7

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

   5. Simplified 92.7%

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

   if -3.9999999999999997e181 < (*.f64 y z) < 1.0000000000000001e172

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

      2. accelerator-lowering-fma.f64 85.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]

   5. Simplified 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -4 \cdot 10^{+181}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot y\\ \mathbf{elif}\;z \cdot y \leq 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-17}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 61000:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.4e-17) t (if (<= t 61000.0) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.4e-17) {
		tmp = t;
	} else if (t <= 61000.0) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-2.4d-17)) then
        tmp = t
    else if (t <= 61000.0d0) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.4e-17) {
		tmp = t;
	} else if (t <= 61000.0) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.4e-17:
		tmp = t
	elif t <= 61000.0:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.4e-17)
		tmp = t;
	elseif (t <= 61000.0)
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.4e-17)
		tmp = t;
	elseif (t <= 61000.0)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.4e-17], t, If[LessEqual[t, 61000.0], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -2.39999999999999986e-17 or 61000 < t

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 59.2%

      \[\leadsto \color{blue}{t} \]

   if -2.39999999999999986e-17 < t < 61000

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 48.9

         \[\leadsto \color{blue}{0.125 \cdot x} \]

   5. Simplified 48.9%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. div-inv N/A

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

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

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

   20. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot -0.5, y, \mathsf{fma}\left(0.125, x, t\right)\right)} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.125, x, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma 0.125 x t))

C

double code(double x, double y, double z, double t) {
	return fma(0.125, x, t);
}

Julia

function code(x, y, z, t)
	return fma(0.125, x, t)
end

Wolfram

code[x_, y_, z_, t_] := N[(0.125 * x + t), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

   2. accelerator-lowering-fma.f64 64.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]

5. Simplified 64.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]
6. Add Preprocessing

Alternative 8: 33.2% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 t)

C

double code(double x, double y, double z, double t) {
	return 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 = t
end function

Java

public static double code(double x, double y, double z, double t) {
	return t;
}

Python

def code(x, y, z, t):
	return t

Julia

function code(x, y, z, t)
	return t
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Simplified 36.2%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (+ (/ x 8.0) t) (* (/ z 2.0) y)))

C

double code(double x, double y, double z, double t) {
	return ((x / 8.0) + t) - ((z / 2.0) * 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 = ((x / 8.0d0) + t) - ((z / 2.0d0) * y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x / 8.0) + t) - ((z / 2.0) * y);
}

Python

def code(x, y, z, t):
	return ((x / 8.0) + t) - ((z / 2.0) * y)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / 8.0) + t) - Float64(Float64(z / 2.0) * y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / 8.0) + t) - ((z / 2.0) * y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / 8.0), $MachinePrecision] + t), $MachinePrecision] - N[(N[(z / 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ (/ x 8) t) (* (/ z 2) y)))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))