Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.1% → 99.8%

Time: 8.6s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, -x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+20)
   (fma (/ y z) x (- x))
   (if (<= z 1.25e+16) (/ (fma (- y z) x x) z) (* (- (/ y z) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+20) {
		tmp = fma((y / z), x, -x);
	} else if (z <= 1.25e+16) {
		tmp = fma((y - z), x, x) / z;
	} else {
		tmp = ((y / z) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+20)
		tmp = fma(Float64(y / z), x, Float64(-x));
	elseif (z <= 1.25e+16)
		tmp = Float64(fma(Float64(y - z), x, x) / z);
	else
		tmp = Float64(Float64(Float64(y / z) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+20], N[(N[(y / z), $MachinePrecision] * x + (-x)), $MachinePrecision], If[LessEqual[z, 1.25e+16], N[(N[(N[(y - z), $MachinePrecision] * x + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e20

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot x + -1 \cdot x} \]

       7. neg-mul-1 N/A

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

       8. lift-neg.f64 N/A

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

       9. lower-fma.f64 99.9

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

   11. Applied rewrites 99.9%

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

   if -2e20 < z < 1.25e16

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.25e16 < z

   1. Initial program 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

      \[\leadsto \left(\color{blue}{\frac{y}{z}} - 1\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \mathsf{fma}\left(-x, z, x\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) - -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e-58)
   (/ (fma y x (fma (- x) z x)) z)
   (/ x (/ z (- (- y z) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e-58) {
		tmp = fma(y, x, fma(-x, z, x)) / z;
	} else {
		tmp = x / (z / ((y - z) - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e-58)
		tmp = Float64(fma(y, x, fma(Float64(-x), z, x)) / z);
	else
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e-58], N[(N[(y * x + N[((-x) * z + x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.0000000000000002e-58

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if 8.0000000000000002e-58 < x

   1. Initial program 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+l- N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. associate--r+ N/A

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

      14. lift--.f64 N/A

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

      15. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) - -1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, -x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z} - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.3)
   (fma (/ y z) x (- x))
   (if (<= z 1.0) (/ (fma y x x) z) (* (- (/ y z) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.3) {
		tmp = fma((y / z), x, -x);
	} else if (z <= 1.0) {
		tmp = fma(y, x, x) / z;
	} else {
		tmp = ((y / z) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.3)
		tmp = fma(Float64(y / z), x, Float64(-x));
	elseif (z <= 1.0)
		tmp = Float64(fma(y, x, x) / z);
	else
		tmp = Float64(Float64(Float64(y / z) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.3], N[(N[(y / z), $MachinePrecision] * x + (-x)), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y * x + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.29999999999999982

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 97.7

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

   9. Applied rewrites 97.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot x + -1 \cdot x} \]

       7. neg-mul-1 N/A

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

       8. lift-neg.f64 N/A

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

       9. lower-fma.f64 97.7

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

   11. Applied rewrites 97.7%

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

   if -5.29999999999999982 < z < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot x + 1 \cdot x}}{z} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{y \cdot x + \color{blue}{x}}{z} \]

      4. lower-fma.f64 98.6

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]

   if 1 < z

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 98.5

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

   9. Applied rewrites 98.5%

      \[\leadsto \left(\color{blue}{\frac{y}{z}} - 1\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y}{z} - 1\right) \cdot x\\ \mathbf{if}\;z \leq -5.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (/ y z) 1.0) x)))
   (if (<= z -5.3) t_0 (if (<= z 1.0) (/ (fma y x x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) - 1.0) * x;
	double tmp;
	if (z <= -5.3) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = fma(y, x, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) - 1.0) * x)
	tmp = 0.0
	if (z <= -5.3)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(fma(y, x, x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -5.3], t$95$0, If[LessEqual[z, 1.0], N[(N[(y * x + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.29999999999999982 or 1 < z

   1. Initial program 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 98.1

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

   9. Applied rewrites 98.1%

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

   if -5.29999999999999982 < z < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot x + 1 \cdot x}}{z} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{y \cdot x + \color{blue}{x}}{z} \]

      4. lower-fma.f64 98.6

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]

   5. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{-45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \mathsf{fma}\left(-x, z, x\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 + y}{z} - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.26e-45)
   (/ (fma y x (fma (- x) z x)) z)
   (* (- (/ (+ 1.0 y) z) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.26e-45) {
		tmp = fma(y, x, fma(-x, z, x)) / z;
	} else {
		tmp = (((1.0 + y) / z) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.26e-45)
		tmp = Float64(fma(y, x, fma(Float64(-x), z, x)) / z);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + y) / z) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.26e-45], N[(N[(y * x + N[((-x) * z + x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.26e-45

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-rgt-identity N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if 1.26e-45 < x

   1. Initial program 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\frac{1 + y}{z} - 1\right)} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500000000000:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 0.0098:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9500000000000.0)
   (* (/ x z) y)
   (if (<= y 0.0098) (- (/ x z) x) (/ (fma y x x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9500000000000.0) {
		tmp = (x / z) * y;
	} else if (y <= 0.0098) {
		tmp = (x / z) - x;
	} else {
		tmp = fma(y, x, x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9500000000000.0)
		tmp = Float64(Float64(x / z) * y);
	elseif (y <= 0.0098)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(fma(y, x, x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9500000000000.0], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 0.0098], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(y * x + x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e12

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      3. lower-/.f64 77.6

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot y \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   if -9.5e12 < y < 0.0097999999999999997

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} + x \cdot -1} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} + x \cdot -1 \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{z} + x \cdot -1 \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{z} + \color{blue}{-1 \cdot x} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      11. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      13. lower-/.f64 100.0

          \[\leadsto \color{blue}{\frac{x}{z}} - x \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   if 0.0097999999999999997 < y

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot x + 1 \cdot x}}{z} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{y \cdot x + \color{blue}{x}}{z} \]

      4. lower-fma.f64 76.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, x\right)}}{z} \]

   5. Applied rewrites 76.2%

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

Alternative 7: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500000000000:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9500000000000.0)
   (* (/ x z) y)
   (if (<= y 2.7e+103) (- (/ x z) x) (/ (* y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9500000000000.0) {
		tmp = (x / z) * y;
	} else if (y <= 2.7e+103) {
		tmp = (x / z) - x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-9500000000000.0d0)) then
        tmp = (x / z) * y
    else if (y <= 2.7d+103) then
        tmp = (x / z) - x
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -9500000000000.0) {
		tmp = (x / z) * y;
	} else if (y <= 2.7e+103) {
		tmp = (x / z) - x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9500000000000.0:
		tmp = (x / z) * y
	elif y <= 2.7e+103:
		tmp = (x / z) - x
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9500000000000.0)
		tmp = Float64(Float64(x / z) * y);
	elseif (y <= 2.7e+103)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -9500000000000.0)
		tmp = (x / z) * y;
	elseif (y <= 2.7e+103)
		tmp = (x / z) - x;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9500000000000.0], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.7e+103], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e12

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      3. lower-/.f64 77.6

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot y \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   if -9.5e12 < y < 2.69999999999999993e103

   1. Initial program 85.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} + x \cdot -1} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} + x \cdot -1 \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{z} + x \cdot -1 \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{z} + \color{blue}{-1 \cdot x} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      11. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      13. lower-/.f64 95.6

          \[\leadsto \color{blue}{\frac{x}{z}} - x \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   if 2.69999999999999993e103 < y

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} \]

      2. lower-*.f64 83.9

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} \]

   5. Applied rewrites 83.9%

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

Alternative 8: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -9500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x z) y)))
   (if (<= y -9500000000000.0) t_0 (if (<= y 5.6e+41) (- (/ x z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) * y;
	double tmp;
	if (y <= -9500000000000.0) {
		tmp = t_0;
	} else if (y <= 5.6e+41) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / z) * y
    if (y <= (-9500000000000.0d0)) then
        tmp = t_0
    else if (y <= 5.6d+41) then
        tmp = (x / z) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) * y;
	double tmp;
	if (y <= -9500000000000.0) {
		tmp = t_0;
	} else if (y <= 5.6e+41) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) * y
	tmp = 0
	if y <= -9500000000000.0:
		tmp = t_0
	elif y <= 5.6e+41:
		tmp = (x / z) - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) * y)
	tmp = 0.0
	if (y <= -9500000000000.0)
		tmp = t_0;
	elseif (y <= 5.6e+41)
		tmp = Float64(Float64(x / z) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) * y;
	tmp = 0.0;
	if (y <= -9500000000000.0)
		tmp = t_0;
	elseif (y <= 5.6e+41)
		tmp = (x / z) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -9500000000000.0], t$95$0, If[LessEqual[y, 5.6e+41], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e12 or 5.5999999999999999e41 < y

   1. Initial program 88.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      3. lower-/.f64 76.1

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot y \]

   5. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   if -9.5e12 < y < 5.5999999999999999e41

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} + x \cdot -1} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} + x \cdot -1 \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{z} + x \cdot -1 \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{z} + \color{blue}{-1 \cdot x} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      11. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      13. lower-/.f64 99.2

          \[\leadsto \color{blue}{\frac{x}{z}} - x \]

   5. Applied rewrites 99.2%

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

Alternative 9: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{-45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 + y}{z} - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.26e-45) (/ (fma (- y z) x x) z) (* (- (/ (+ 1.0 y) z) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.26e-45) {
		tmp = fma((y - z), x, x) / z;
	} else {
		tmp = (((1.0 + y) / z) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.26e-45)
		tmp = Float64(fma(Float64(y - z), x, x) / z);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + y) / z) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.26e-45], N[(N[(N[(y - z), $MachinePrecision] * x + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.26e-45

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if 1.26e-45 < x

   1. Initial program 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\frac{1 + y}{z} - 1\right)} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (- x) (if (<= z 1.0) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = -x
    else if (z <= 1.0d0) then
        tmp = x / z
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -x;
	} else if (z <= 1.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -x
	elif z <= 1.0:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(-x);
	elseif (z <= 1.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = -x;
	elseif (z <= 1.0)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], (-x), If[LessEqual[z, 1.0], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 74.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 74.3

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{-x} \]

   if -1 < z < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{z} + x \cdot -1} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} + x \cdot -1 \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{z} + x \cdot -1 \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{z} + \color{blue}{-1 \cdot x} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      11. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{z} - x} \]

      13. lower-/.f64 57.3

          \[\leadsto \color{blue}{\frac{x}{z}} - x \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.2%

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

Alternative 11: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} - x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ x z) x))

C

double code(double x, double y, double z) {
	return (x / z) - x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / z) - x
end function

Java

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

Python

def code(x, y, z):
	return (x / z) - x

Julia

function code(x, y, z)
	return Float64(Float64(x / z) - x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{z}} \]

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. *-inverses N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{z} + x \cdot -1} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{x \cdot 1}{z}} + x \cdot -1 \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{x}}{z} + x \cdot -1 \]

   9. *-commutative N/A

      \[\leadsto \frac{x}{z} + \color{blue}{-1 \cdot x} \]

   10. mul-1-neg N/A

       \[\leadsto \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   11. unsub-neg N/A

       \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   12. lower--.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   13. lower-/.f64 66.6

       \[\leadsto \color{blue}{\frac{x}{z}} - x \]

5. Applied rewrites 66.6%

   \[\leadsto \color{blue}{\frac{x}{z} - x} \]
6. Add Preprocessing

Alternative 12: 40.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 38.2

      \[\leadsto \color{blue}{-x} \]

5. Applied rewrites 38.2%

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

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024242 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

  (/ (* x (+ (- y z) 1.0)) z))