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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

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, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-0.5 * z), $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. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 2: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -2e-13) (not (<= t_1 2e-11)))
     (fma -0.5 (* z y) t)
     (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if ((t_1 <= -2e-13) || !(t_1 <= 2e-11)) {
		tmp = fma(-0.5, (z * y), t);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if ((t_1 <= -2e-13) || !(t_1 <= 2e-11))
		tmp = fma(-0.5, Float64(z * y), t);
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-13], N[Not[LessEqual[t$95$1, 2e-11]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision] + t), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -2.0000000000000001e-13 or 1.99999999999999988e-11 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if -2.0000000000000001e-13 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 1.99999999999999988e-11

   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. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -2 \cdot 10^{-13} \lor \neg \left(\frac{y \cdot z}{2} \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (<= t_1 -2e-13)
     (fma (* -0.5 z) y t)
     (if (<= t_1 2e-11) (fma 0.125 x t) (fma -0.5 (* z y) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = fma((-0.5 * z), y, t);
	} else if (t_1 <= 2e-11) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = fma(-0.5, (z * y), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if (t_1 <= -2e-13)
		tmp = fma(Float64(-0.5 * z), y, t);
	elseif (t_1 <= 2e-11)
		tmp = fma(0.125, x, t);
	else
		tmp = fma(-0.5, Float64(z * y), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-13], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(0.125 * x + t), $MachinePrecision], N[(-0.5 * N[(z * y), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -2.0000000000000001e-13

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 84.2%

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

   if -2.0000000000000001e-13 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 1.99999999999999988e-11

   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. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if 1.99999999999999988e-11 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

Alternative 4: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot z}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+104} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+115}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* y z) 2.0)))
   (if (or (<= t_1 -2e+104) (not (<= t_1 2e+115)))
     (* -0.5 (* z y))
     (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) / 2.0;
	double tmp;
	if ((t_1 <= -2e+104) || !(t_1 <= 2e+115)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) / 2.0)
	tmp = 0.0
	if ((t_1 <= -2e+104) || !(t_1 <= 2e+115))
		tmp = Float64(-0.5 * Float64(z * y));
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+104], N[Not[LessEqual[t$95$1, 2e+115]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y z) #s(literal 2 binary64)) < -2e104 or 2e115 < (/.f64 (*.f64 y z) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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. lower-fma.f64 16.2

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 82.6%

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

   if -2e104 < (/.f64 (*.f64 y z) #s(literal 2 binary64)) < 2e115

   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. lower-fma.f64 86.5

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

   5. Applied rewrites 86.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot z}{2} \leq -2 \cdot 10^{+104} \lor \neg \left(\frac{y \cdot z}{2} \leq 2 \cdot 10^{+115}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+20} \lor \neg \left(x \leq 8.5 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(0.125, x, -0.5 \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.6e+20) (not (<= x 8.5e+19)))
   (fma 0.125 x (* -0.5 (* y z)))
   (fma (* -0.5 z) y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.6e+20) || !(x <= 8.5e+19)) {
		tmp = fma(0.125, x, (-0.5 * (y * z)));
	} else {
		tmp = fma((-0.5 * z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.6e+20) || !(x <= 8.5e+19))
		tmp = fma(0.125, x, Float64(-0.5 * Float64(y * z)));
	else
		tmp = fma(Float64(-0.5 * z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.6e+20], N[Not[LessEqual[x, 8.5e+19]], $MachinePrecision]], N[(0.125 * x + N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.6e20 or 8.5e19 < x

   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. lower-fma.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -8.6e20 < x < 8.5e19

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+20} \lor \neg \left(x \leq 8.5 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(0.125, x, -0.5 \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, x \cdot 0.125\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, -0.5 \cdot \left(y \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -8.6e+20)
   (fma (* -0.5 z) y (* x 0.125))
   (if (<= x 8.5e+19) (fma (* -0.5 z) y t) (fma 0.125 x (* -0.5 (* y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -8.6e+20) {
		tmp = fma((-0.5 * z), y, (x * 0.125));
	} else if (x <= 8.5e+19) {
		tmp = fma((-0.5 * z), y, t);
	} else {
		tmp = fma(0.125, x, (-0.5 * (y * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -8.6e+20)
		tmp = fma(Float64(-0.5 * z), y, Float64(x * 0.125));
	elseif (x <= 8.5e+19)
		tmp = fma(Float64(-0.5 * z), y, t);
	else
		tmp = fma(0.125, x, Float64(-0.5 * Float64(y * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -8.6e+20], N[(N[(-0.5 * z), $MachinePrecision] * y + N[(x * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+19], N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision], N[(0.125 * x + N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.6e20

   1. Initial program 99.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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. lower-fma.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   10. Applied rewrites 86.1%

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

   if -8.6e20 < x < 8.5e19

   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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.5%

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

   if 8.5e19 < x

   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. lower-fma.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in t around 0

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

      1. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 90.3

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

   8. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, -0.5 \cdot \left(y \cdot z\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(-0.5 * N[(z * y), $MachinePrecision] + 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. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 8: 63.5% 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. lower-fma.f64 62.6

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

5. Applied rewrites 62.6%

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

Alternative 9: 32.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot x \end{array} \]

FPCore

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

C

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

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 = 0.125d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.125 * x;
}

Python

def code(x, y, z, t):
	return 0.125 * x

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.125 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.125 * x), $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. lower-fma.f64 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in t around 0

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

   1. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 68.5

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

8. Applied rewrites 68.5%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 32.1

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

11. Applied rewrites 32.1%

    \[\leadsto \color{blue}{0.125 \cdot x} \]
12. 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 2024338 
(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))