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

Percentage Accurate: 88.3% → 93.7%

Time: 5.8s

Alternatives: 6

Speedup: 0.7×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

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: 88.3% 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: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+196}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e+196)
   (- x)
   (if (<= z 1.16e+114) (/ (fma x (- y z) x) z) (- (/ x z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+196) {
		tmp = -x;
	} else if (z <= 1.16e+114) {
		tmp = fma(x, (y - z), x) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.6e+196)
		tmp = Float64(-x);
	elseif (z <= 1.16e+114)
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.6e+196], (-x), If[LessEqual[z, 1.16e+114], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999961e196

   1. Initial program 63.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 95.7

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

   5. Simplified 95.7%

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

   if -4.59999999999999961e196 < z < 1.15999999999999994e114

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.0

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

   5. Simplified 96.0%

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

   if 1.15999999999999994e114 < z

   1. Initial program 58.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 78.9

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

   5. Simplified 78.9%

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

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+80}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4e+80) (- x) (if (<= z 2.95e+60) (/ (fma x y x) z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4e+80) {
		tmp = -x;
	} else if (z <= 2.95e+60) {
		tmp = fma(x, y, x) / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4e+80)
		tmp = Float64(-x);
	elseif (z <= 2.95e+60)
		tmp = Float64(fma(x, y, x) / z);
	else
		tmp = Float64(-x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4e+80], (-x), If[LessEqual[z, 2.95e+60], N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -4e80 or 2.9500000000000001e60 < z

   1. Initial program 69.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 77.1

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

   5. Simplified 77.1%

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

   if -4e80 < z < 2.9500000000000001e60

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. *-inverses N/A

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

      15. associate-*r* N/A

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

      16. *-rgt-identity N/A

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

      17. +-commutative N/A

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

      18. distribute-lft1-in N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 92.5

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

   5. Simplified 92.5%

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

Alternative 3: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= y -5.5e+58) t_0 (if (<= y 5.5e+44) (- (/ x z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if (y <= -5.5e+58) {
		tmp = t_0;
	} else if (y <= 5.5e+44) {
		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 * y) / z
    if (y <= (-5.5d+58)) then
        tmp = t_0
    else if (y <= 5.5d+44) 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 * y) / z;
	double tmp;
	if (y <= -5.5e+58) {
		tmp = t_0;
	} else if (y <= 5.5e+44) {
		tmp = (x / z) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if y <= -5.5e+58:
		tmp = t_0
	elif y <= 5.5e+44:
		tmp = (x / z) - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (y <= -5.5e+58)
		tmp = t_0;
	elseif (y <= 5.5e+44)
		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 * y) / z;
	tmp = 0.0;
	if (y <= -5.5e+58)
		tmp = t_0;
	elseif (y <= 5.5e+44)
		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 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -5.5e+58], t$95$0, If[LessEqual[y, 5.5e+44], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999999e58 or 5.5000000000000001e44 < y

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

   if -5.4999999999999999e58 < y < 5.5000000000000001e44

   1. Initial program 86.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 94.5

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

   5. Simplified 94.5%

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

Alternative 4: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3350000000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 45000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3350000000000.0) (- x) (if (<= z 45000.0) (/ x z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3350000000000.0) {
		tmp = -x;
	} else if (z <= 45000.0) {
		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 <= (-3350000000000.0d0)) then
        tmp = -x
    else if (z <= 45000.0d0) 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 <= -3350000000000.0) {
		tmp = -x;
	} else if (z <= 45000.0) {
		tmp = x / z;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3350000000000.0:
		tmp = -x
	elif z <= 45000.0:
		tmp = x / z
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3350000000000.0)
		tmp = Float64(-x);
	elseif (z <= 45000.0)
		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 <= -3350000000000.0)
		tmp = -x;
	elseif (z <= 45000.0)
		tmp = x / z;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3350000000000.0], (-x), If[LessEqual[z, 45000.0], N[(x / z), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.35e12 or 45000 < z

   1. Initial program 75.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 67.1

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

   5. Simplified 67.1%

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

   if -3.35e12 < z < 45000

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 51.2

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

   5. Simplified 51.2%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 50.7

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

   8. Simplified 50.7%

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

Alternative 5: 66.4% 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 86.7%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 59.6

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

5. Simplified 59.6%

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

Alternative 6: 38.8% 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 86.7%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 37.0

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

5. Simplified 37.0%

   \[\leadsto \color{blue}{-x} \]
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 2024215 
(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))