Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.9% → 99.6%

Time: 2.7s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(a, \frac{z - y}{1 + \left(t - z\right)}, x\right) \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (fma a (/ (- z y) (+ 1.0 (- t z))) x))

C

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

Julia

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

Wolfram

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

ReFlow

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

Derivation

1. Initial program 96.9%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Step-by-step derivation

3. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{1 + \left(t - z\right)}, x\right) \]
4. Add Preprocessing

Alternative 2: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -9.209060127716842 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{t}, x\right)\\ \mathbf{elif}\;t \leq 1.588974140430518 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{t}{a}}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= t -9.209060127716842e+100)
  (fma a (/ (- z y) t) x)
  (if (<= t 1.588974140430518e+72)
    (fma a (/ (- z y) (- 1.0 z)) x)
    (- x (/ (- y z) (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.209060127716842e+100) {
		tmp = fma(a, ((z - y) / t), x);
	} else if (t <= 1.588974140430518e+72) {
		tmp = fma(a, ((z - y) / (1.0 - z)), x);
	} else {
		tmp = x - ((y - z) / (t / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.209060127716842e+100)
		tmp = fma(a, Float64(Float64(z - y) / t), x);
	elseif (t <= 1.588974140430518e+72)
		tmp = fma(a, Float64(Float64(z - y) / Float64(1.0 - z)), x);
	else
		tmp = Float64(x - Float64(Float64(y - z) / Float64(t / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.209060127716842e+100], N[(a * N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.588974140430518e+72], N[(a * N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y - z), $MachinePrecision] / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_1 = IF (t <= (1588974140430517904977279078016891919585776927988155899991536063735660544)) THEN ((a * ((z - y) / ((1) - z))) + x) ELSE (x - ((y - z) / (t / a))) ENDIF IN
	LET tmp = IF (t <= (-92090601277168424412061736530652869730985372047551950660329594233465220283737972226271163357119643648)) THEN ((a * ((z - y) / t)) + x) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if t < -9.2090601277168424e100

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.8%

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

   if -9.2090601277168424e100 < t < 1.5889741404305179e72

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{1 - z}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{1 - z}, x\right) \]

   if 1.5889741404305179e72 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.3%

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

Alternative 3: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{a}{z}, x\right)\\ \mathbf{if}\;z \leq -429393245428857860:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9669578702788 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{a \cdot y}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (let* ((t_1 (fma (- y z) (/ a z) x)))
  (if (<= z -429393245428857860.0)
    t_1
    (if (<= z 2.9669578702788e-8) (- x (/ (* a y) (+ 1.0 t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (a / z), x);
	double tmp;
	if (z <= -429393245428857860.0) {
		tmp = t_1;
	} else if (z <= 2.9669578702788e-8) {
		tmp = x - ((a * y) / (1.0 + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(a / z), x)
	tmp = 0.0
	if (z <= -429393245428857860.0)
		tmp = t_1;
	elseif (z <= 2.9669578702788e-8)
		tmp = Float64(x - Float64(Float64(a * y) / Float64(1.0 + t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(a / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -429393245428857860.0], t$95$1, If[LessEqual[z, 2.9669578702788e-8], N[(x - N[(N[(a * y), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET t_1 = (((y - z) * (a / z)) + x) IN
		LET tmp_1 = IF (z <= (2966957870278799904431975427952783430640693040913902223110198974609375e-77)) THEN (x - ((a * y) / ((1) + t))) ELSE t_1 ENDIF IN
		LET tmp = IF (z <= (-429393245428857856)) THEN t_1 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if z < -429393245428857860 or 2.9669578702787999e-8 < z

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{\left(z - 1\right) - t}, x\right) \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{z}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 57.1%

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

   if -429393245428857860 < z < 2.9669578702787999e-8

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto x - \frac{a \cdot y}{1 + t} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.8%

      \[\leadsto x - \frac{a \cdot y}{1 + t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -8.743689116393911 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{t}, x\right)\\ \mathbf{elif}\;t \leq -8.48977571085483 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{elif}\;t \leq 1.588974140430518 \cdot 10^{+72}:\\ \;\;\;\;x - a \cdot \frac{y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= t -8.743689116393911e+100)
  (fma a (/ (- z y) t) x)
  (if (<= t -8.48977571085483e-164)
    (fma a (/ z (- 1.0 z)) x)
    (if (<= t 1.588974140430518e+72)
      (- x (* a (/ y (- 1.0 z))))
      (- x (* a (/ y t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.743689116393911e+100) {
		tmp = fma(a, ((z - y) / t), x);
	} else if (t <= -8.48977571085483e-164) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else if (t <= 1.588974140430518e+72) {
		tmp = x - (a * (y / (1.0 - z)));
	} else {
		tmp = x - (a * (y / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.743689116393911e+100)
		tmp = fma(a, Float64(Float64(z - y) / t), x);
	elseif (t <= -8.48977571085483e-164)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	elseif (t <= 1.588974140430518e+72)
		tmp = Float64(x - Float64(a * Float64(y / Float64(1.0 - z))));
	else
		tmp = Float64(x - Float64(a * Float64(y / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.743689116393911e+100], N[(a * N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, -8.48977571085483e-164], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.588974140430518e+72], N[(x - N[(a * N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(a * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_2 = IF (t <= (1588974140430517904977279078016891919585776927988155899991536063735660544)) THEN (x - (a * (y / ((1) - z)))) ELSE (x - (a * (y / t))) ENDIF IN
	LET tmp_1 = IF (t <= (-848977571085482954365607604259322329905206278806212609486785644141778936815181015554688261768445364966165597293041854026566030599358464187466693859079982044539480619304055445241599751228790727372864694252566680182456945199224774707318769082811469138592612369872451032475711589319712134597267186857973982053174639508356301721526938855810399541857152049987851877725345773930485010594468482658658814443697337992489337921142578125e-589)) THEN ((a * (z / ((1) - z))) + x) ELSE tmp_2 ENDIF IN
	LET tmp = IF (t <= (-87436891163939109535450069906425861963793904862902628923866482999012006214380180979616755430698516480)) THEN ((a * ((z - y) / t)) + x) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 4 regimes

2. if t < -8.743689116393911e100

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.8%

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

   if -8.743689116393911e100 < t < -8.4897757108548295e-164

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]

   if -8.4897757108548295e-164 < t < 1.5889741404305179e72

   1. Initial program 96.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\left(1 + t\right) - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 64.5%

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

   if 1.5889741404305179e72 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto x - \frac{a \cdot y}{t} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.7%

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

Alternative 5: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{a}{z}, x\right)\\ \mathbf{if}\;z \leq -261491804051375550:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1689062569814557 \cdot 10^{-136}:\\ \;\;\;\;x - \frac{a \cdot y}{1}\\ \mathbf{elif}\;z \leq 7.89290889567554 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 + t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (let* ((t_1 (fma (- y z) (/ a z) x)))
  (if (<= z -261491804051375550.0)
    t_1
    (if (<= z 1.1689062569814557e-136)
      (- x (/ (* a y) 1.0))
      (if (<= z 7.89290889567554e-12)
        (fma a (/ z (+ 1.0 t)) x)
        t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (a / z), x);
	double tmp;
	if (z <= -261491804051375550.0) {
		tmp = t_1;
	} else if (z <= 1.1689062569814557e-136) {
		tmp = x - ((a * y) / 1.0);
	} else if (z <= 7.89290889567554e-12) {
		tmp = fma(a, (z / (1.0 + t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(a / z), x)
	tmp = 0.0
	if (z <= -261491804051375550.0)
		tmp = t_1;
	elseif (z <= 1.1689062569814557e-136)
		tmp = Float64(x - Float64(Float64(a * y) / 1.0));
	elseif (z <= 7.89290889567554e-12)
		tmp = fma(a, Float64(z / Float64(1.0 + t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(a / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -261491804051375550.0], t$95$1, If[LessEqual[z, 1.1689062569814557e-136], N[(x - N[(N[(a * y), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.89290889567554e-12], N[(a * N[(z / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET t_1 = (((y - z) * (a / z)) + x) IN
		LET tmp_2 = IF (z <= (789290889567554046096926610693745988457659112924602595739997923374176025390625e-89)) THEN ((a * (z / ((1) + t))) + x) ELSE t_1 ENDIF IN
		LET tmp_1 = IF (z <= (11689062569814557456504976687070482409168411299026905829703030298874197016597937994529717240947662636646126623515081481690239314632902078667122720135730772354784079318280137075426975073255251841757348857709604822537724875013433797741375912953717502583566997403752518111418666854064383539062009774271516212773705533754777790710240037430622805914026685059070587158203125e-503)) THEN (x - ((a * y) / (1))) ELSE tmp_2 ENDIF IN
		LET tmp = IF (z <= (-261491804051375552)) THEN t_1 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if z < -261491804051375550 or 7.8929088956755405e-12 < z

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{\left(z - 1\right) - t}, x\right) \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{z}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 57.1%

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

   if -261491804051375550 < z < 1.1689062569814557e-136

   1. Initial program 96.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\left(1 + t\right) - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in z around 0

      \[\leadsto x - \frac{a \cdot y}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.7%

       \[\leadsto x - \frac{a \cdot y}{1} \]

   if 1.1689062569814557e-136 < z < 7.8929088956755405e-12

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + t}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.0%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + t}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -8.743689116393911 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{t}, x\right)\\ \mathbf{elif}\;t \leq 1.9958465885143322 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= t -8.743689116393911e+100)
  (fma a (/ (- z y) t) x)
  (if (<= t 1.9958465885143322e-46)
    (fma a (/ z (- 1.0 z)) x)
    (- x (* a (/ y t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.743689116393911e+100) {
		tmp = fma(a, ((z - y) / t), x);
	} else if (t <= 1.9958465885143322e-46) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else {
		tmp = x - (a * (y / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.743689116393911e+100)
		tmp = fma(a, Float64(Float64(z - y) / t), x);
	elseif (t <= 1.9958465885143322e-46)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	else
		tmp = Float64(x - Float64(a * Float64(y / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.743689116393911e+100], N[(a * N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.9958465885143322e-46], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(a * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_1 = IF (t <= (19958465885143322253071183511186748514139706798809594708584493125509105130665800245771396699095847904254686631353682239875535486817170749418437480926513671875e-203)) THEN ((a * (z / ((1) - z))) + x) ELSE (x - (a * (y / t))) ENDIF IN
	LET tmp = IF (t <= (-87436891163939109535450069906425861963793904862902628923866482999012006214380180979616755430698516480)) THEN ((a * ((z - y) / t)) + x) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if t < -8.743689116393911e100

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 53.8%

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

   if -8.743689116393911e100 < t < 1.9958465885143322e-46

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]

   if 1.9958465885143322e-46 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto x - \frac{a \cdot y}{t} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.7%

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

Alternative 7: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2401274959286454 \cdot 10^{+69}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.3771913992391488 \cdot 10^{-136}:\\ \;\;\;\;x - \frac{a \cdot y}{1}\\ \mathbf{elif}\;z \leq 51415045947863040:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= z -5.2401274959286454e+69)
  (- x a)
  (if (<= z 1.3771913992391488e-136)
    (- x (/ (* a y) 1.0))
    (if (<= z 51415045947863040.0) (fma a (/ z 1.0) x) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2401274959286454e+69) {
		tmp = x - a;
	} else if (z <= 1.3771913992391488e-136) {
		tmp = x - ((a * y) / 1.0);
	} else if (z <= 51415045947863040.0) {
		tmp = fma(a, (z / 1.0), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2401274959286454e+69)
		tmp = Float64(x - a);
	elseif (z <= 1.3771913992391488e-136)
		tmp = Float64(x - Float64(Float64(a * y) / 1.0));
	elseif (z <= 51415045947863040.0)
		tmp = fma(a, Float64(z / 1.0), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2401274959286454e+69], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.3771913992391488e-136], N[(x - N[(N[(a * y), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 51415045947863040.0], N[(a * N[(z / 1.0), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_2 = IF (z <= (51415045947863040)) THEN ((a * (z / (1))) + x) ELSE (x - a) ENDIF IN
	LET tmp_1 = IF (z <= (137719139923914883681944525330971478511046996123033030587505701586521985420841293423718472004529110964270902096520263061753749232379440214459689109088594308037818205638525007326136094654167066970417654116480605388773610279843571087779173264007168996963436338110461591300116299672566806084041454815907436380935623330127370446776015333600895473864511586725711822509765625e-504)) THEN (x - ((a * y) / (1))) ELSE tmp_2 ENDIF IN
	LET tmp = IF (z <= (-5240127495928645404650702366282045090084663310735805993680651404967936)) THEN (x - a) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if z < -5.2401274959286454e69 or 51415045947863040 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x - a \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

      \[\leadsto x - a \]

   if -5.2401274959286454e69 < z < 1.3771913992391488e-136

   1. Initial program 96.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\left(1 + t\right) - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in z around 0

      \[\leadsto x - \frac{a \cdot y}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.7%

       \[\leadsto x - \frac{a \cdot y}{1} \]

   if 1.3771913992391488e-136 < z < 51415045947863040

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]

   10. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(a, \frac{z}{1}, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 43.2%

       \[\leadsto \mathsf{fma}\left(a, \frac{z}{1}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -7.100329979148691 \cdot 10^{+82}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5.096755374494824 \cdot 10^{-12}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= z -7.100329979148691e+82)
  (- x a)
  (if (<= z 5.096755374494824e-12) (- x (* a (/ y t))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.100329979148691e+82) {
		tmp = x - a;
	} else if (z <= 5.096755374494824e-12) {
		tmp = x - (a * (y / t));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-7.100329979148691d+82)) then
        tmp = x - a
    else if (z <= 5.096755374494824d-12) then
        tmp = x - (a * (y / t))
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.100329979148691e+82) {
		tmp = x - a;
	} else if (z <= 5.096755374494824e-12) {
		tmp = x - (a * (y / t));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.100329979148691e+82:
		tmp = x - a
	elif z <= 5.096755374494824e-12:
		tmp = x - (a * (y / t))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.100329979148691e+82)
		tmp = Float64(x - a);
	elseif (z <= 5.096755374494824e-12)
		tmp = Float64(x - Float64(a * Float64(y / t)));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.100329979148691e+82)
		tmp = x - a;
	elseif (z <= 5.096755374494824e-12)
		tmp = x - (a * (y / t));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.100329979148691e+82], N[(x - a), $MachinePrecision], If[LessEqual[z, 5.096755374494824e-12], N[(x - N[(a * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_1 = IF (z <= (5096755374494823614096771616268003574599931848609912776737473905086517333984375e-90)) THEN (x - (a * (y / t))) ELSE (x - a) ENDIF IN
	LET tmp = IF (z <= (-71003299791486909743695491055666916355018625220269571925477465805918605213993795584)) THEN (x - a) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if z < -7.100329979148691e82 or 5.0967553744948236e-12 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x - a \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

      \[\leadsto x - a \]

   if -7.100329979148691e82 < z < 5.0967553744948236e-12

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto x - \frac{a \cdot y}{t} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.7%

      \[\leadsto x - a \cdot \frac{y}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -8089.580774177239:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 51415045947863040:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (if (<= z -8089.580774177239)
  (- x a)
  (if (<= z 51415045947863040.0) (fma a (/ z 1.0) x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8089.580774177239) {
		tmp = x - a;
	} else if (z <= 51415045947863040.0) {
		tmp = fma(a, (z / 1.0), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8089.580774177239)
		tmp = Float64(x - a);
	elseif (z <= 51415045947863040.0)
		tmp = fma(a, Float64(z / 1.0), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8089.580774177239], N[(x - a), $MachinePrecision], If[LessEqual[z, 51415045947863040.0], N[(a * N[(z / 1.0), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET tmp_1 = IF (z <= (51415045947863040)) THEN ((a * (z / (1))) + x) ELSE (x - a) ENDIF IN
	LET tmp = IF (z <= (-8089580774177238708944059908390045166015625e-39)) THEN (x - a) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if z < -8089.5807741772387 or 51415045947863040 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x - a \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

      \[\leadsto x - a \]

   if -8089.5807741772387 < z < 51415045947863040

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 + \left(t - z\right)}, x\right) \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]

   10. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(a, \frac{z}{1}, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 43.2%

       \[\leadsto \mathsf{fma}\left(a, \frac{z}{1}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{a \cdot y}{-1}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+250}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (let* ((t_1 (/ (* a y) -1.0)) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
  (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 2e+250) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * y) / -1.0;
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+250) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * y) / -1.0;
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+250) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * y) / -1.0
	t_2 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+250:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * y) / -1.0)
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+250)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * y) / -1.0;
	t_2 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+250)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * y), $MachinePrecision] / -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+250], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -inf.0 or 1.9999999999999998e250 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{\left(z - 1\right) - t}, x\right) \]

   4. Taylor expanded in y around inf

      \[\leadsto \frac{a \cdot y}{z - \left(1 + t\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 24.6%

      \[\leadsto \frac{a \cdot y}{z - \left(1 + t\right)} \]

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 18.0%

      \[\leadsto \frac{a \cdot y}{z - 1} \]

   10. Taylor expanded in z around 0

       \[\leadsto \frac{a \cdot y}{-1} \]
   11. Step-by-step derivation

   12. Applied rewrites 14.9%

       \[\leadsto \frac{a \cdot y}{-1} \]

   if -inf.0 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.9999999999999998e250

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x - a \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

Alternative 11: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := a \cdot \frac{y}{z}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+250}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  :pre TRUE
  (let* ((t_1 (* a (/ y z))) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
  (if (<= t_2 -5e+284) t_1 (if (<= t_2 2e+250) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / z);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -5e+284) {
		tmp = t_1;
	} else if (t_2 <= 2e+250) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a * (y / z)
    t_2 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_2 <= (-5d+284)) then
        tmp = t_1
    else if (t_2 <= 2d+250) then
        tmp = x - a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / z);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -5e+284) {
		tmp = t_1;
	} else if (t_2 <= 2e+250) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * (y / z)
	t_2 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_2 <= -5e+284:
		tmp = t_1
	elif t_2 <= 2e+250:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * Float64(y / z))
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -5e+284)
		tmp = t_1;
	elseif (t_2 <= 2e+250)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * (y / z);
	t_2 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_2 <= -5e+284)
		tmp = t_1;
	elseif (t_2 <= 2e+250)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+284], t$95$1, If[LessEqual[t$95$2, 2e+250], N[(x - a), $MachinePrecision], t$95$1]]]]

ReFlow

f(x, y, z, t, a):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a: real): real =
	LET t_1 = (a * (y / z)) IN
		LET t_2 = ((y - z) / (((t - z) + (1)) / a)) IN
			LET tmp_1 = IF (t_2 <= (19999999999999998421936661664294053151080855387504444745733035393436825233278672005560948283410708288220728162236284648020809571429082630568562515505514583247285006834145935719548240949300738322281106670384019261349564171109391944306795105153030553600)) THEN (x - a) ELSE t_1 ENDIF IN
			LET tmp = IF (t_2 <= (-499999999999999990079578960260221425096554759926423605900012855280825179991269042612044308093073246922143074698610725186309660447719446848973826083227612667029686373206874073603221710445876270310293765181110136931503450775547995353849221420762954771236422294344040540188066309300289536)) THEN t_1 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -4.9999999999999999e284 or 1.9999999999999998e250 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

   3. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(y - z, \frac{a}{\left(z - 1\right) - t}, x\right) \]

   4. Taylor expanded in y around inf

      \[\leadsto \frac{a \cdot y}{z - \left(1 + t\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 24.6%

      \[\leadsto \frac{a \cdot y}{z - \left(1 + t\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{z} \]
   8. Step-by-step derivation

   9. Applied rewrites 8.9%

      \[\leadsto \frac{a \cdot y}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 10.1%

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

   if -4.9999999999999999e284 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.9999999999999998e250

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto x - a \]
   3. Step-by-step derivation

   4. Applied rewrites 58.9%

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

Alternative 12: 58.9% accurate, 5.3× speedup

Math

\[x - a \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

ReFlow

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

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in z around inf

   \[\leadsto x - a \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

   \[\leadsto x - a \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))