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

Percentage Accurate: 88.5% → 99.5%

Time: 7.5s

Alternatives: 7

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: 88.5% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 10^{-260}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-292)
     t_0
     (if (<= t_0 1e-260) (- (- z) (/ (* z (+ x z)) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-292) {
		tmp = t_0;
	} else if (t_0 <= 1e-260) {
		tmp = -z - ((z * (x + z)) / y);
	} 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) / (1.0d0 - (y / z))
    if (t_0 <= (-5d-292)) then
        tmp = t_0
    else if (t_0 <= 1d-260) then
        tmp = -z - ((z * (x + z)) / y)
    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) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-292) {
		tmp = t_0;
	} else if (t_0 <= 1e-260) {
		tmp = -z - ((z * (x + z)) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-292:
		tmp = t_0
	elif t_0 <= 1e-260:
		tmp = -z - ((z * (x + z)) / y)
	else:
		tmp = t_0
	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-292)
		tmp = t_0;
	elseif (t_0 <= 1e-260)
		tmp = Float64(Float64(-z) - Float64(Float64(z * Float64(x + z)) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -5e-292)
		tmp = t_0;
	elseif (t_0 <= 1e-260)
		tmp = -z - ((z * (x + z)) / y);
	else
		tmp = t_0;
	end
	tmp_2 = 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[LessEqual[t$95$0, -5e-292], t$95$0, If[LessEqual[t$95$0, 1e-260], N[((-z) - N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999981e-292 or 9.99999999999999961e-261 < (/.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 -4.99999999999999981e-292 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 9.99999999999999961e-261

   1. Initial program 14.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. div-sub N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. distribute-rgt-out N/A

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

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

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

      19. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 10^{-260}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z - y}\\ \mathbf{if}\;y \leq -48000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y (- z y)))))
   (if (<= y -48000.0)
     t_0
     (if (<= y 1.8e-118)
       (+ x y)
       (if (<= y 1.3e+31) (* x (/ z (- z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / (z - y));
	double tmp;
	if (y <= -48000.0) {
		tmp = t_0;
	} else if (y <= 1.8e-118) {
		tmp = x + y;
	} else if (y <= 1.3e+31) {
		tmp = x * (z / (z - y));
	} 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 = z * (y / (z - y))
    if (y <= (-48000.0d0)) then
        tmp = t_0
    else if (y <= 1.8d-118) then
        tmp = x + y
    else if (y <= 1.3d+31) then
        tmp = x * (z / (z - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y / (z - y));
	double tmp;
	if (y <= -48000.0) {
		tmp = t_0;
	} else if (y <= 1.8e-118) {
		tmp = x + y;
	} else if (y <= 1.3e+31) {
		tmp = x * (z / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / (z - y))
	tmp = 0
	if y <= -48000.0:
		tmp = t_0
	elif y <= 1.8e-118:
		tmp = x + y
	elif y <= 1.3e+31:
		tmp = x * (z / (z - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / Float64(z - y)))
	tmp = 0.0
	if (y <= -48000.0)
		tmp = t_0;
	elseif (y <= 1.8e-118)
		tmp = Float64(x + y);
	elseif (y <= 1.3e+31)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / (z - y));
	tmp = 0.0;
	if (y <= -48000.0)
		tmp = t_0;
	elseif (y <= 1.8e-118)
		tmp = x + y;
	elseif (y <= 1.3e+31)
		tmp = x * (z / (z - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -48000.0], t$95$0, If[LessEqual[y, 1.8e-118], N[(x + y), $MachinePrecision], If[LessEqual[y, 1.3e+31], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -48000 or 1.3e31 < y

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 83.8

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

   5. Simplified 83.8%

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

   if -48000 < y < 1.8000000000000001e-118

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 86.3

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

   5. Simplified 86.3%

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

   if 1.8000000000000001e-118 < y < 1.3e31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 58.7

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

   5. Simplified 58.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 66.9

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

   7. Applied egg-rr 66.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -48000:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-118}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{if}\;y \leq -2600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-55}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma z (/ x y) z))))
   (if (<= y -2600000.0) t_0 (if (<= y 5.3e-55) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -fma(z, (x / y), z);
	double tmp;
	if (y <= -2600000.0) {
		tmp = t_0;
	} else if (y <= 5.3e-55) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(z, Float64(x / y), z))
	tmp = 0.0
	if (y <= -2600000.0)
		tmp = t_0;
	elseif (y <= 5.3e-55)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -2600000.0], t$95$0, If[LessEqual[y, 5.3e-55], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e6 or 5.3000000000000003e-55 < y

   1. Initial program 72.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

      23. neg-lowering-neg.f64 N/A

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

   5. Simplified 74.8%

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

   if -2.6e6 < y < 5.3000000000000003e-55

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 83.4

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

   5. Simplified 83.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2600000:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-55}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+76}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+136) (- z) (if (<= y 5.1e+76) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+136) {
		tmp = -z;
	} else if (y <= 5.1e+76) {
		tmp = x + y;
	} else {
		tmp = -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 <= (-2.4d+136)) then
        tmp = -z
    else if (y <= 5.1d+76) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+136) {
		tmp = -z;
	} else if (y <= 5.1e+76) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+136:
		tmp = -z
	elif y <= 5.1e+76:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+136)
		tmp = Float64(-z);
	elseif (y <= 5.1e+76)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+136)
		tmp = -z;
	elseif (y <= 5.1e+76)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+136], (-z), If[LessEqual[y, 5.1e+76], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e136 or 5.1000000000000002e76 < y

   1. Initial program 61.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. neg-lowering-neg.f64 79.6

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

   5. Simplified 79.6%

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

   if -2.4e136 < y < 5.1000000000000002e76

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 73.1

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

   5. Simplified 73.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+76}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.235:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 0.00057:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.235) (- z) (if (<= y 0.00057) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.235) {
		tmp = -z;
	} else if (y <= 0.00057) {
		tmp = x;
	} else {
		tmp = -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 <= (-0.235d0)) then
        tmp = -z
    else if (y <= 0.00057d0) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.235) {
		tmp = -z;
	} else if (y <= 0.00057) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.235:
		tmp = -z
	elif y <= 0.00057:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.235)
		tmp = Float64(-z);
	elseif (y <= 0.00057)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.235)
		tmp = -z;
	elseif (y <= 0.00057)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.235], (-z), If[LessEqual[y, 0.00057], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -0.23499999999999999 or 5.6999999999999998e-4 < y

   1. Initial program 71.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. neg-lowering-neg.f64 65.7

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

   5. Simplified 65.7%

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

   if -0.23499999999999999 < y < 5.6999999999999998e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 65.0%

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

Alternative 6: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-194}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-190}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.4e-194) x (if (<= x 4.5e-190) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e-194) {
		tmp = x;
	} else if (x <= 4.5e-190) {
		tmp = y;
	} 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 (x <= (-5.4d-194)) then
        tmp = x
    else if (x <= 4.5d-190) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e-194) {
		tmp = x;
	} else if (x <= 4.5e-190) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.4e-194:
		tmp = x
	elif x <= 4.5e-190:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.4e-194)
		tmp = x;
	elseif (x <= 4.5e-190)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.4e-194)
		tmp = x;
	elseif (x <= 4.5e-190)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.4e-194], x, If[LessEqual[x, 4.5e-190], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e-194 or 4.50000000000000021e-190 < x

   1. Initial program 86.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 47.0%

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

   if -5.4e-194 < x < 4.50000000000000021e-190

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 42.9%

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

Alternative 7: 34.8% accurate, 29.0× 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 x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

5. Simplified 40.1%

   \[\leadsto \color{blue}{x} \]
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 2024205 
(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))))