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

Percentage Accurate: 88.0% → 99.8%

Time: 3.5s

Alternatives: 8

Speedup: 0.7×

Specification

Math

\[\frac{x + y}{1 - \frac{y}{z}} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (/ (+ 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x + y) / ((1) - (y / z))
END code

Initial Program: 88.0% accurate, 1.0× speedup

Math

\[\frac{x + y}{1 - \frac{y}{z}} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (/ (+ 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x + y) / ((1) - (y / z))
END code

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -3.782608026078261 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{\frac{z - y}{y + x}}\\ \mathbf{elif}\;y \leq 2.782787727800268 \cdot 10^{+40}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y + x}{z - y}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= y -3.782608026078261e+52)
  (/ z (/ (- z y) (+ y x)))
  (if (<= y 2.782787727800268e+40)
    (/ (+ x y) (- 1.0 (/ y z)))
    (* z (/ (+ y x) (- z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.782608026078261e+52) {
		tmp = z / ((z - y) / (y + x));
	} else if (y <= 2.782787727800268e+40) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = z * ((y + x) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.782608026078261d+52)) then
        tmp = z / ((z - y) / (y + x))
    else if (y <= 2.782787727800268d+40) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = z * ((y + x) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.782608026078261e+52) {
		tmp = z / ((z - y) / (y + x));
	} else if (y <= 2.782787727800268e+40) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = z * ((y + x) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.782608026078261e+52:
		tmp = z / ((z - y) / (y + x))
	elif y <= 2.782787727800268e+40:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = z * ((y + x) / (z - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.782608026078261e+52)
		tmp = Float64(z / Float64(Float64(z - y) / Float64(y + x)));
	elseif (y <= 2.782787727800268e+40)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(z * Float64(Float64(y + x) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.782608026078261e+52)
		tmp = z / ((z - y) / (y + x));
	elseif (y <= 2.782787727800268e+40)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = z * ((y + x) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.782608026078261e+52], N[(z / N[(N[(z - y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.782787727800268e+40], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_1 = IF (y <= (27827877278002678491622181790330216513536)) THEN ((x + y) / ((1) - (y / z))) ELSE (z * ((y + x) / (z - y))) ENDIF IN
	LET tmp = IF (y <= (-37826080260782611780427230706470323719289090349203456)) THEN (z / ((z - y) / (y + x))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if y < -3.7826080260782612e52

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   if -3.7826080260782612e52 < y < 2.7827877278002678e40

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   if 2.7827877278002678e40 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 92.3%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -4.779069952465816 \cdot 10^{+41}:\\ \;\;\;\;\frac{z}{\frac{z - y}{y + x}}\\ \mathbf{elif}\;y \leq 2.782787727800268 \cdot 10^{+40}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y + x}{z - y}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= y -4.779069952465816e+41)
  (/ z (/ (- z y) (+ y x)))
  (if (<= y 2.782787727800268e+40)
    (* (+ y x) (/ z (- z y)))
    (* z (/ (+ y x) (- z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.779069952465816e+41) {
		tmp = z / ((z - y) / (y + x));
	} else if (y <= 2.782787727800268e+40) {
		tmp = (y + x) * (z / (z - y));
	} else {
		tmp = z * ((y + x) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.779069952465816d+41)) then
        tmp = z / ((z - y) / (y + x))
    else if (y <= 2.782787727800268d+40) then
        tmp = (y + x) * (z / (z - y))
    else
        tmp = z * ((y + x) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.779069952465816e+41) {
		tmp = z / ((z - y) / (y + x));
	} else if (y <= 2.782787727800268e+40) {
		tmp = (y + x) * (z / (z - y));
	} else {
		tmp = z * ((y + x) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.779069952465816e+41:
		tmp = z / ((z - y) / (y + x))
	elif y <= 2.782787727800268e+40:
		tmp = (y + x) * (z / (z - y))
	else:
		tmp = z * ((y + x) / (z - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.779069952465816e+41)
		tmp = Float64(z / Float64(Float64(z - y) / Float64(y + x)));
	elseif (y <= 2.782787727800268e+40)
		tmp = Float64(Float64(y + x) * Float64(z / Float64(z - y)));
	else
		tmp = Float64(z * Float64(Float64(y + x) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.779069952465816e+41)
		tmp = z / ((z - y) / (y + x));
	elseif (y <= 2.782787727800268e+40)
		tmp = (y + x) * (z / (z - y));
	else
		tmp = z * ((y + x) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.779069952465816e+41], N[(z / N[(N[(z - y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.782787727800268e+40], N[(N[(y + x), $MachinePrecision] * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_1 = IF (y <= (27827877278002678491622181790330216513536)) THEN ((y + x) * (z / (z - y))) ELSE (z * ((y + x) / (z - y))) ENDIF IN
	LET tmp = IF (y <= (-477906995246581564721202183051219172327424)) THEN (z / ((z - y) / (y + x))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if y < -4.7790699524658156e41

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   if -4.7790699524658156e41 < y < 2.7827877278002678e40

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   if 2.7827877278002678e40 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 92.3%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := z \cdot \frac{y + x}{z - y}\\ \mathbf{if}\;y \leq -2.349987207420596 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.782787727800268 \cdot 10^{+40}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* z (/ (+ y x) (- z y)))))
  (if (<= y -2.349987207420596e+38)
    t_0
    (if (<= y 2.782787727800268e+40) (* (+ y x) (/ z (- z y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((y + x) / (z - y));
	double tmp;
	if (y <= -2.349987207420596e+38) {
		tmp = t_0;
	} else if (y <= 2.782787727800268e+40) {
		tmp = (y + x) * (z / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 + x) / (z - y))
    if (y <= (-2.349987207420596d+38)) then
        tmp = t_0
    else if (y <= 2.782787727800268d+40) then
        tmp = (y + 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 + x) / (z - y));
	double tmp;
	if (y <= -2.349987207420596e+38) {
		tmp = t_0;
	} else if (y <= 2.782787727800268e+40) {
		tmp = (y + x) * (z / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((y + x) / (z - y))
	tmp = 0
	if y <= -2.349987207420596e+38:
		tmp = t_0
	elif y <= 2.782787727800268e+40:
		tmp = (y + x) * (z / (z - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(y + x) / Float64(z - y)))
	tmp = 0.0
	if (y <= -2.349987207420596e+38)
		tmp = t_0;
	elseif (y <= 2.782787727800268e+40)
		tmp = Float64(Float64(y + 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 + x) / (z - y));
	tmp = 0.0;
	if (y <= -2.349987207420596e+38)
		tmp = t_0;
	elseif (y <= 2.782787727800268e+40)
		tmp = (y + 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[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.349987207420596e+38], t$95$0, If[LessEqual[y, 2.782787727800268e+40], N[(N[(y + x), $MachinePrecision] * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (z * ((y + x) / (z - y))) IN
		LET tmp_1 = IF (y <= (27827877278002678491622181790330216513536)) THEN ((y + x) * (z / (z - y))) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-234998720742059599760376513388584370176)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -2.349987207420596e38 or 2.7827877278002678e40 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 92.3%

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

   if -2.349987207420596e38 < y < 2.7827877278002678e40

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

Alternative 4: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := z \cdot \frac{y + x}{z - y}\\ \mathbf{if}\;y \leq -1.393380278335505 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2590884414245868 \cdot 10^{-121}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* z (/ (+ y x) (- z y)))))
  (if (<= y -1.393380278335505e-132)
    t_0
    (if (<= y 1.2590884414245868e-121) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((y + x) / (z - y));
	double tmp;
	if (y <= -1.393380278335505e-132) {
		tmp = t_0;
	} else if (y <= 1.2590884414245868e-121) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 + x) / (z - y))
    if (y <= (-1.393380278335505d-132)) then
        tmp = t_0
    else if (y <= 1.2590884414245868d-121) then
        tmp = x + 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 + x) / (z - y));
	double tmp;
	if (y <= -1.393380278335505e-132) {
		tmp = t_0;
	} else if (y <= 1.2590884414245868e-121) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((y + x) / (z - y))
	tmp = 0
	if y <= -1.393380278335505e-132:
		tmp = t_0
	elif y <= 1.2590884414245868e-121:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(y + x) / Float64(z - y)))
	tmp = 0.0
	if (y <= -1.393380278335505e-132)
		tmp = t_0;
	elseif (y <= 1.2590884414245868e-121)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((y + x) / (z - y));
	tmp = 0.0;
	if (y <= -1.393380278335505e-132)
		tmp = t_0;
	elseif (y <= 1.2590884414245868e-121)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.393380278335505e-132], t$95$0, If[LessEqual[y, 1.2590884414245868e-121], N[(x + y), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (z * ((y + x) / (z - y))) IN
		LET tmp_1 = IF (y <= (12590884414245867916964251781670553425417828492279866593888021910993250080090161244863213897198858140141885910576060382130240866304915151059412932972558710258646141320215692889407817085927726543953633644674597646893887990114242500969246794803344558933696047747674296307472368679095407872762547862866000514259212650358676910400390625e-452)) THEN (x + y) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-139338027833550495421609237833855785807980735895209602394223859933222193787226464124841903037081523468805133003768223682553246453031977759931545536458730632658134641763278300327297454251337311798227706847060284131383382562262257891156626774541822376601076696158484243728964800060628392342937379722448231107140534479706985016012055211831466294825077056884765625e-491)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -1.393380278335505e-132 or 1.2590884414245868e-121 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 92.3%

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

   if -1.393380278335505e-132 < y < 1.2590884414245868e-121

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + y \]
   10. Step-by-step derivation

   11. Applied rewrites 50.6%

       \[\leadsto x + y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{y}{z - y} \cdot z\\ \mathbf{if}\;y \leq -3.2190743546567084 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.73466363203287 \cdot 10^{-181}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 1.218357395521096 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (/ y (- z y)) z)))
  (if (<= y -3.2190743546567084e-42)
    t_0
    (if (<= y -4.73466363203287e-181)
      (* x (/ z (- z y)))
      (if (<= y 1.218357395521096e-7) (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (y / (z - y)) * z;
	double tmp;
	if (y <= -3.2190743546567084e-42) {
		tmp = t_0;
	} else if (y <= -4.73466363203287e-181) {
		tmp = x * (z / (z - y));
	} else if (y <= 1.218357395521096e-7) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 / (z - y)) * z
    if (y <= (-3.2190743546567084d-42)) then
        tmp = t_0
    else if (y <= (-4.73466363203287d-181)) then
        tmp = x * (z / (z - y))
    else if (y <= 1.218357395521096d-7) then
        tmp = x + 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 = (y / (z - y)) * z;
	double tmp;
	if (y <= -3.2190743546567084e-42) {
		tmp = t_0;
	} else if (y <= -4.73466363203287e-181) {
		tmp = x * (z / (z - y));
	} else if (y <= 1.218357395521096e-7) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / (z - y)) * z
	tmp = 0
	if y <= -3.2190743546567084e-42:
		tmp = t_0
	elif y <= -4.73466363203287e-181:
		tmp = x * (z / (z - y))
	elif y <= 1.218357395521096e-7:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(z - y)) * z)
	tmp = 0.0
	if (y <= -3.2190743546567084e-42)
		tmp = t_0;
	elseif (y <= -4.73466363203287e-181)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	elseif (y <= 1.218357395521096e-7)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / (z - y)) * z;
	tmp = 0.0;
	if (y <= -3.2190743546567084e-42)
		tmp = t_0;
	elseif (y <= -4.73466363203287e-181)
		tmp = x * (z / (z - y));
	elseif (y <= 1.218357395521096e-7)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -3.2190743546567084e-42], t$95$0, If[LessEqual[y, -4.73466363203287e-181], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.218357395521096e-7], N[(x + y), $MachinePrecision], t$95$0]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = ((y / (z - y)) * z) IN
		LET tmp_2 = IF (y <= (121835739552109587254476760591337392014565921272151172161102294921875e-75)) THEN (x + y) ELSE t_0 ENDIF IN
		LET tmp_1 = IF (y <= (-4734663632032869917938737926775996076304363301531924046625645605168081914623443707539272548654688762786328737096233062537483599475836292175195387950595192747046857263437979868700255469409554212834747288476811883514543915831968463510360557257527096592511330425377670450249512797864810003927783307768410582167096042022980766605779449371542008021586849605881386176575670165330092589403825487986570776211361618841752453989202417675347979442168622199460514821112155914306640625e-652)) THEN (x * (z / (z - y))) ELSE tmp_2 ENDIF IN
		LET tmp = IF (y <= (-321907435465670843865925379648631418932056154541407274247954021849173747255142939905268844251927711768855762397567588095625978894531726837158203125e-188)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if y < -3.2190743546567084e-42 or 1.2183573955210959e-7 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 42.0%

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

   6. Applied rewrites 49.8%

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

   if -3.2190743546567084e-42 < y < -4.7346636320328699e-181

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 48.5%

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

   if -4.7346636320328699e-181 < y < 1.2183573955210959e-7

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + y \]
   10. Step-by-step derivation

   11. Applied rewrites 50.6%

       \[\leadsto x + y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{z}{z - y}\\ \mathbf{if}\;y \leq -8.03123504347913 \cdot 10^{+206}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.2190743546567084 \cdot 10^{-42}:\\ \;\;\;\;y \cdot t\_0\\ \mathbf{elif}\;y \leq -4.73466363203287 \cdot 10^{-181}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{elif}\;y \leq 4.681666818528092 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ z (- z y))))
  (if (<= y -8.03123504347913e+206)
    (- z)
    (if (<= y -3.2190743546567084e-42)
      (* y t_0)
      (if (<= y -4.73466363203287e-181)
        (* x t_0)
        (if (<= y 4.681666818528092e+63) (+ x y) (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = z / (z - y);
	double tmp;
	if (y <= -8.03123504347913e+206) {
		tmp = -z;
	} else if (y <= -3.2190743546567084e-42) {
		tmp = y * t_0;
	} else if (y <= -4.73466363203287e-181) {
		tmp = x * t_0;
	} else if (y <= 4.681666818528092e+63) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 / (z - y)
    if (y <= (-8.03123504347913d+206)) then
        tmp = -z
    else if (y <= (-3.2190743546567084d-42)) then
        tmp = y * t_0
    else if (y <= (-4.73466363203287d-181)) then
        tmp = x * t_0
    else if (y <= 4.681666818528092d+63) 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 t_0 = z / (z - y);
	double tmp;
	if (y <= -8.03123504347913e+206) {
		tmp = -z;
	} else if (y <= -3.2190743546567084e-42) {
		tmp = y * t_0;
	} else if (y <= -4.73466363203287e-181) {
		tmp = x * t_0;
	} else if (y <= 4.681666818528092e+63) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z / (z - y)
	tmp = 0
	if y <= -8.03123504347913e+206:
		tmp = -z
	elif y <= -3.2190743546567084e-42:
		tmp = y * t_0
	elif y <= -4.73466363203287e-181:
		tmp = x * t_0
	elif y <= 4.681666818528092e+63:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z / Float64(z - y))
	tmp = 0.0
	if (y <= -8.03123504347913e+206)
		tmp = Float64(-z);
	elseif (y <= -3.2190743546567084e-42)
		tmp = Float64(y * t_0);
	elseif (y <= -4.73466363203287e-181)
		tmp = Float64(x * t_0);
	elseif (y <= 4.681666818528092e+63)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z / (z - y);
	tmp = 0.0;
	if (y <= -8.03123504347913e+206)
		tmp = -z;
	elseif (y <= -3.2190743546567084e-42)
		tmp = y * t_0;
	elseif (y <= -4.73466363203287e-181)
		tmp = x * t_0;
	elseif (y <= 4.681666818528092e+63)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.03123504347913e+206], (-z), If[LessEqual[y, -3.2190743546567084e-42], N[(y * t$95$0), $MachinePrecision], If[LessEqual[y, -4.73466363203287e-181], N[(x * t$95$0), $MachinePrecision], If[LessEqual[y, 4.681666818528092e+63], N[(x + y), $MachinePrecision], (-z)]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (z / (z - y)) IN
		LET tmp_3 = IF (y <= (4681666818528092203036118676955019348905995956188990931967410176)) THEN (x + y) ELSE (- z) ENDIF IN
		LET tmp_2 = IF (y <= (-4734663632032869917938737926775996076304363301531924046625645605168081914623443707539272548654688762786328737096233062537483599475836292175195387950595192747046857263437979868700255469409554212834747288476811883514543915831968463510360557257527096592511330425377670450249512797864810003927783307768410582167096042022980766605779449371542008021586849605881386176575670165330092589403825487986570776211361618841752453989202417675347979442168622199460514821112155914306640625e-652)) THEN (x * t_0) ELSE tmp_3 ENDIF IN
		LET tmp_1 = IF (y <= (-321907435465670843865925379648631418932056154541407274247954021849173747255142939905268844251927711768855762397567588095625978894531726837158203125e-188)) THEN (y * t_0) ELSE tmp_2 ENDIF IN
		LET tmp = IF (y <= (-803123504347913009913273388224513530471665723338522923978604741586461544313617260628683085621740752238018556851792543899478292393703587484937124172655362631178881111759137227320373164508270778886514532155392)) THEN (- z) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 4 regimes

2. if y < -8.0312350434791301e206 or 4.6816668185280922e63 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf

      \[\leadsto -1 \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 34.9%

      \[\leadsto -1 \cdot z \]
   5. Step-by-step derivation

   6. Applied rewrites 34.9%

      \[\leadsto -z \]

   if -8.0312350434791301e206 < y < -3.2190743546567084e-42

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 42.0%

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

   6. Applied rewrites 42.6%

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

   if -3.2190743546567084e-42 < y < -4.7346636320328699e-181

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 48.5%

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

   if -4.7346636320328699e-181 < y < 4.6816668185280922e63

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + y \]
   10. Step-by-step derivation

   11. Applied rewrites 50.6%

       \[\leadsto x + y \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -6.632586809221022 \cdot 10^{+115}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.681666818528092 \cdot 10^{+63}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (if (<= y -6.632586809221022e+115)
  (- z)
  (if (<= y 4.681666818528092e+63) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.632586809221022e+115) {
		tmp = -z;
	} else if (y <= 4.681666818528092e+63) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6.632586809221022d+115)) then
        tmp = -z
    else if (y <= 4.681666818528092d+63) 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 <= -6.632586809221022e+115) {
		tmp = -z;
	} else if (y <= 4.681666818528092e+63) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.632586809221022e+115:
		tmp = -z
	elif y <= 4.681666818528092e+63:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.632586809221022e+115)
		tmp = Float64(-z);
	elseif (y <= 4.681666818528092e+63)
		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 <= -6.632586809221022e+115)
		tmp = -z;
	elseif (y <= 4.681666818528092e+63)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.632586809221022e+115], (-z), If[LessEqual[y, 4.681666818528092e+63], N[(x + y), $MachinePrecision], (-z)]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_1 = IF (y <= (4681666818528092203036118676955019348905995956188990931967410176)) THEN (x + y) ELSE (- z) ENDIF IN
	LET tmp = IF (y <= (-66325868092210219984679563846771621306077676758794470155515335874769546430828771989845968860035693982987589091590144)) THEN (- z) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -6.632586809221022e115 or 4.6816668185280922e63 < y

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf

      \[\leadsto -1 \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 34.9%

      \[\leadsto -1 \cdot z \]
   5. Step-by-step derivation

   6. Applied rewrites 34.9%

      \[\leadsto -z \]

   if -6.632586809221022e115 < y < 4.6816668185280922e63

   1. Initial program 88.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 92.4%

      \[\leadsto \frac{z}{\frac{z - y}{y + x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + y \]
   10. Step-by-step derivation

   11. Applied rewrites 50.6%

       \[\leadsto x + y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 34.9% accurate, 6.9× speedup

Math

\[-z \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    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)

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	- z
END code

Derivation

1. Initial program 88.0%

   \[\frac{x + y}{1 - \frac{y}{z}} \]

2. Taylor expanded in y around inf

   \[\leadsto -1 \cdot z \]
3. Step-by-step derivation

4. Applied rewrites 34.9%

   \[\leadsto -1 \cdot z \]
5. Step-by-step derivation

6. Applied rewrites 34.9%

   \[\leadsto -z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64
  (/ (+ x y) (- 1.0 (/ y z))))