Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.2% → 99.6%

Time: 6.6s

Alternatives: 5

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / 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) / (1.0d0 - (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / 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) / (1.0d0 - (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-266} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y} + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-266) (not (<= t_0 0.0)))
     t_0
     (- (+ (/ (fma z x (* z z)) y) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-266) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -((fma(z, x, (z * z)) / y) + z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-266) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-Float64(Float64(fma(z, x, Float64(z * z)) / y) + z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-266], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(N[(N[(z * x + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999992e-266 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -4.99999999999999992e-266 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 12.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 10.0%

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

   6. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + -1 \cdot z} \]

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-266} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y} + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-6} \lor \neg \left(y \leq 4.5 \cdot 10^{+77}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.1e-6) (not (<= y 4.5e+77))) (* z (- -1.0 (/ x y))) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e-6) || !(y <= 4.5e+77)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = y + 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 ((y <= (-1.1d-6)) .or. (.not. (y <= 4.5d+77))) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e-6) || !(y <= 4.5e+77)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.1e-6) or not (y <= 4.5e+77):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.1e-6) || !(y <= 4.5e+77))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.1e-6) || ~((y <= 4.5e+77)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.1e-6], N[Not[LessEqual[y, 4.5e+77]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1000000000000001e-6 or 4.50000000000000024e77 < y

   1. Initial program 74.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-add N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. *-commutative N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

   5. Applied rewrites 79.4%

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

   if -1.1000000000000001e-6 < y < 4.50000000000000024e77

   1. Initial program 99.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 79.4

         \[\leadsto \color{blue}{y + x} \]

   8. Applied rewrites 79.4%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-6} \lor \neg \left(y \leq 4.5 \cdot 10^{+77}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{z}{y}, z, z\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+77}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+113) (- (fma (/ z y) z z)) (if (<= y 5.5e+77) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+113) {
		tmp = -fma((z / y), z, z);
	} else if (y <= 5.5e+77) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+113)
		tmp = Float64(-fma(Float64(z / y), z, z));
	elseif (y <= 5.5e+77)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+113], (-N[(N[(z / y), $MachinePrecision] * z + z), $MachinePrecision]), If[LessEqual[y, 5.5e+77], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999984e113

   1. Initial program 68.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

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

         \[\leadsto -1 \cdot z - \frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y} \]

      9. *-lft-identity N/A

         \[\leadsto -1 \cdot z - \color{blue}{1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

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

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

      11. metadata-eval N/A

          \[\leadsto -1 \cdot z + \color{blue}{-1} \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} \]

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto -\left(z + \frac{{z}^{2}}{y}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.8%

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

   if -7.19999999999999984e113 < y < 5.50000000000000036e77

   1. Initial program 98.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 75.4

         \[\leadsto \color{blue}{y + x} \]

   8. Applied rewrites 75.4%

      \[\leadsto \color{blue}{y + x} \]

   if 5.50000000000000036e77 < y

   1. Initial program 72.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.3

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

   5. Applied rewrites 64.3%

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

Alternative 4: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+113} \lor \neg \left(y \leq 5.5 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.2e+113) (not (<= y 5.5e+77))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.2e+113) || !(y <= 5.5e+77)) {
		tmp = -z;
	} else {
		tmp = y + 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 ((y <= (-7.2d+113)) .or. (.not. (y <= 5.5d+77))) then
        tmp = -z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.2e+113) || !(y <= 5.5e+77)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.2e+113) or not (y <= 5.5e+77):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.2e+113) || !(y <= 5.5e+77))
		tmp = Float64(-z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.2e+113) || ~((y <= 5.5e+77)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.2e+113], N[Not[LessEqual[y, 5.5e+77]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999984e113 or 5.50000000000000036e77 < y

   1. Initial program 70.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -7.19999999999999984e113 < y < 5.50000000000000036e77

   1. Initial program 98.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 75.4

         \[\leadsto \color{blue}{y + x} \]

   8. Applied rewrites 75.4%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+113} \lor \neg \left(y \leq 5.5 \cdot 10^{+77}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 33.8

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

5. Applied rewrites 33.8%

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

Developer Target 1: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / 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 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / 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 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

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